Abstract
Let \((X_n)_{n\geqslant 0}\) be a Markov chain with values in a finite state space \({\mathbb {X}}\) starting at \(X_0=x \in {\mathbb {X}}\) and let f be a real function defined on \({\mathbb {X}}\). Set \(S_n=\sum _{k=1}^{n} f(X_k)\), \(n\geqslant 1\). For any \(y \in {\mathbb {R}}\) denote by \(\tau _y\) the first time when \(y+S_n\) becomes non-positive. We study the asymptotic behaviour of the probability \({\mathbb {P}}_x \left( y+S_{n} \in [z,z+a],\, \tau _y > n \right) \) as \(n\rightarrow +\infty .\) We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order \(n^{3/2}\) and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability \({\mathbb {P}}_x \left( \tau _y = n \right) \) as \(n\rightarrow +\infty \).
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1 Introduction
Assume that on the probability space \((\varOmega , {\mathscr {F}}, {\mathbb {P}})\) we are given a sequence of real valued random variables \(( X_n )_{n\geqslant 1}\). Consider the random walk \(S_n=\sum _{k=1}^{n}X_k\), \(n\geqslant 1.\) Suppose first that \(( X_n )_{n\geqslant 1}\) are independent identically distributed of zero mean and finite variance. For any \(y>0\) denote by \(\tau _y\) the first time when \(y+S_n\) becomes non-positive. The study of the asymptotic behaviour of the probability \({\mathbb {P}}(\tau _y>n)\) and of the law of \(y+S_n\) conditioned to stay positive (i.e. given the event \(\{\tau _y>n\}\)) has been initiated by Spitzer [31] and developed subsequently by Iglehart [20], Bolthausen [2], Doney [10], Bertoin and Doney [1], Borovkov [3, 4], to cite only a few. Important progress has been achieved recently by employing a new approach based on the existence of the harmonic function by Denisov and Wachtel [6,7,8] (see also Varopoulos [33, 34] and Eichelbacher and König [11]). In this line Grama, Le Page and Peigné [17] and the authors in [13, 14] have studied sums of functions defined on Markov chains under spectral gap assumptions. The goal of the present paper is to complete these investigations by establishing local limit theorems for random walks defined on finite Markov chains and conditioned to stay positive.
Local limit theorems for the sums of independent random variables without conditioning have attracted much attention, since the pioneering work of Gnedenko [12] and Stone [32]. The first local limit theorem for a random walk conditioned to stay positive has been established in Iglehart [21] in the context of walks with negative drift \({\mathbb {E}} X_1 < 0\). Caravenna [5] studied conditioned local limit theorems for random variables in the domain of attraction of the normal law and Vatutin and Wachtel [35] for random variables \(X_k\) in the domain of attraction of the stable law. Denisov and Wachtel [8] obtained a local limit theorem for random walks in \({\mathbb {Z}}^d\) conditioned to stay in a cone based on the harmonic function approach.
Local limit theorems without conditioning for Markov chains are known as early as the work of Kolmogorov [24] and the background contributions due to Nagaev [27, 28] who initiated the study of Markov chains by spectral methods. The work of Doeblin–Fortet [9] and the theorem of Ionescu-Tulcea and Marinescu [22] allowed to weaken Nagaev’s conditions to deal with Markov kernels having a contraction property. In this spirit Le Page [25] proved a local limit theorem for products of random matrices and Guivarc’h and Hardy [18] and Hennion and Hervé [19] obtained local limit theorems for sums \(S_n=\sum _{k=1}^{n} f(X_k)\), where \(( X_n )_{n\geqslant 0}\) is a Markov chain and f a real function defined on the state space of the chain, which will be also the setting of our paper.
Much less is known on the conditioned local limit theorems. We are aware only of the results of Presman [29, 30] who has considered the case of finite Markov chains in a more general setting but which, because of rather stringent assumptions, do not cover the results of this paper. We can note also the work of Le Page and Peigné [26] where a conditioned local limit theorem is established for the stochastic recursion in a rather different setting.
Let us briefly review the main results of the paper concerning conditioned local limit behaviour of the walk \(S_n=\sum _{k=1}^{n} f(X_k)\) defined on a finite Markov chain \(( X_n )_{n\geqslant 0}\). From more general statement of Theorem 2.4, under the conditions that the underlying Markov chain is irreducible and aperiodic and that \(( S_n )_{n\geqslant 0}\) is centred and non-lattice, for fixed \(x\in {\mathbb {X}}\) and \(y\in {\mathbb {R}}\), it follows that, uniformly in \(z \geqslant 0,\)
where \(\varphi _+(t) = te^{-\frac{t^2}{2}} \mathbb {1}_{\{t\geqslant 0\}}\) is the Rayleigh density. The relation (1.1) is an extension of the classical local limit theorem by Stone [32] to the case of Markov chains. We refer to Caravenna [5] and Vatutin and Wachtel [35], where the corresponding results have been obtained for independent random variables in the domains of attraction of the normal and stable law respectively.
We note that while (1.1) is consistent for large z, it is not informative for z in a compact set. A meaningful local limit behaviour for fixed values of z can be obtained from our Theorem 2.5. Under the same assumptions, for any fixed \(x\in {\mathbb {X}}\), \(y\in {\mathbb {R}}\) and \(z \geqslant 0,\)
For sums of independent random variables similar limit behaviour was found in Vatutin and Wachtel [35]. It should be noted that (1.1) and (1.2) complement each other: the main term in (1.1) is meaningful for large z such that \(z \sim n^{1/2} \) as \(n\rightarrow \infty \), while (1.2) holds for z in compact sets.
We also state extensions of (1.1) and (1.2) to the joint law of \(X_n\) and \(y+S_n\). These extensions are useful in applications, in particular, for determining the exact asymptotic behaviour of the survival time for branching processes in a Markovian environment. They also allow us to infer the local limit behaviour of the exit time \(\tau _y\) (see Theorem 2.8): under the assumptions mentioned before, for any \(x\in {\mathbb {X}}\) and \(y \in {\mathbb {R}}\),
The approach employed in this paper is different from that in [26, 29, 30] which all are based on Wiener-Hopf arguments. Our technique is close to that in Denisov and Wachtel [8], however, in order to make it work for the random walk \(S_n=\sum _{k=1}^{n} f(X_k)\) defined on the Markov chain \(( X_n )_{n\geqslant 0}\), we have to overcome some essential difficulties. One of them is related to the problem of the reversibility of the Markov walk \((S_n )_{n\geqslant 0}\). Let us explain this point in more details. When \(( X_n )_{n\geqslant 1}\) are \({\mathbb {Z}}\)-valued independent identically distributed random variables, let \(( S_n^* )_{n\geqslant 1}\) be the reversed walk given by \(S_n^* = \sum _{k=1}^n X_k^*\), where \(( X_n^* )_{n\geqslant 1}\) is a sequence of independent identically distributed random variables of the same law as \(-X_1\). Denote by \(\tau ^*_z\) the first time when \(( z+S_k^* )_{k\geqslant 0}\) becomes non-positive. Then, due to exchangeability of the random variables \(( X_n )_{n\geqslant 1}\), we have
This relation does not hold any more for the walk \(S_n=\sum _{k=1}^{n} f(X_k)\), where \(( X_n )_{n\geqslant 0}\) is a Markov chain. Even though \(( X_n )_{n\geqslant 0}\) takes values on a finite state space \({\mathbb {X}}\) and there exists a dual chain \(( X^*_n )_{n\geqslant 0},\) the main difficulty is that the function \(f:{\mathbb {X}} \mapsto {\mathbb {R}}\) can be arbitrary and therefore the Markov walk \((S_n )_{n\geqslant 0}\) is not necessarily lattice valued. In this case the Markov chain formed by the couple \(( X_n, y+S_n )_{n\geqslant 0}\) cannot be reversed directly as in (1.3). We cope with this by altering the arrival interval \([z,z+h]\) in the following two-sided bound
where \({\varvec{\nu }}\) is the invariant probability of the Markov chain \(( X_n )_{n\geqslant 1}\), \(\psi _{x}^*: {\mathbb {X}} \mapsto {\mathbb {R}}_+\) is a function such that \({\varvec{\nu }} \left( \psi _{x}^* \right) = 1\) (see (6.2) for a precise definition) and \(S_n^* = -\sum _{k=1}^n f \left( X_k^* \right) \), \(\forall n \geqslant 1.\) Following this idea, for a fixed \(a >0\) we split the interval \([z,z+a]\) into p subintervals of length \(h=a/p\) and we determine the exact upper and lower bounds for the corresponding expectations in (1.4). We then patch up the obtained bounds to obtain a precise asymptotic as \(n \rightarrow +\infty \) for the probabilities \({\mathbb {P}}_x (y+S_n\in [z,z+a], \tau _y > n)\) for a fixed \(a>0\) and let then p go to \(+\infty \). This resumes very succinctly how we suggest generalizing (1.3) to the non-lattice case. Together with some further developments in Sects. 7 and 8, this allows us to establish Theorems 2.4 and 2.5.
The outline of the paper is as follows:
Section 2: We give the necessary notations and formulate the main results.
Section 3: Introduce the dual Markov chain and state some of its properties.
Section 4: Introduce and study the perturbed transition operator.
Section 5: We prove a local limit theorem for sums defined on Markov chains.
Section 6: We collect some auxiliary bounds.
Sections 7, 8 and 9: Proofs of Theorems 2.4, 2.5 and 2.7, 2.8, respectively.
Section 10: We state auxiliary assertions which are necessary for the proofs.
Let us end this section by fixing some notations. The symbol c will denote a positive constant depending on the all previously introduced constants. Sometimes, to stress the dependence of the constants on some parameters \(\alpha ,\beta ,\dots \) we shall use the notations \( c_{\alpha }, c_{\alpha ,\beta },\dots \). All these constants are likely to change their values every occurrence. The indicator of an event A is denoted by \(\mathbb {1}_A\). For any bounded measurable function f on \({\mathbb {X}}\), random variable X in \({\mathbb {X}}\) and event A, the integral \(\int _{{\mathbb {X}}} f(x) {\mathbb {P}} (X \in \text {d}x, A)\) means the expectation \({\mathbb {E}}\left( f(X); A\right) ={\mathbb {E}} \left( f(X) \mathbb {1}_A\right) \).
2 Notations and results
Let \(( X_n )_{n\geqslant 0}\) be a homogeneous Markov chain on the probability space \((\varOmega , {\mathscr {F}}, {\mathbb {P}})\) with values in the finite state space \({\mathbb {X}}\). Denote by \({\mathscr {C}}\) the set of complex functions defined on \({\mathbb {X}}\) endowed with the norm \(\left\| \cdot \right\| _{\infty }\): \(\left\| g\right\| _{\infty } = \sup _{x\in {\mathbb {X}}} \left|g(x)\right|\), for any \(g\in {\mathscr {C}}\). Let \({\mathbf {P}}\) be the transition kernel of the Markov chain \(( X_n )_{n\geqslant 0}\) to which we associate the following transition operator: for any \(x\in {\mathbb {X}}\) and \(g \in {\mathscr {C}}\),
For any \(x\in {\mathbb {X}}\), denote by \({\mathbb {P}}_x\) and \({\mathbb {E}}_x\) the probability, respectively the expectation, generated by the finite dimensional distributions of the Markov chain \(( X_n )_{n\geqslant 0}\) starting at \(X_0 = x\). We assume that the Markov chain is irreducible and aperiodic, which is equivalent to the following hypothesis.
Hypothesis M1
The matrix \({\mathbf {P}}\) is primitive: there exists \(k_0 \geqslant 1\) such that for any \(x \in {\mathbb {X}}\) and any non-negative and non identically zero function \(g \in {\mathscr {C}}\),
Let f be a real valued function defined on \({\mathbb {X}}\) and let \((S_n)_{n\geqslant 0}\) be the process defined by
For any starting point \(y \in {\mathbb {R}}\) we consider the Markov walk \((y+S_n)_{n\geqslant 0}\) and we denote by \(\tau _y\) the first time when the Markov walk becomes non-positive:
Under M1, by the Perron–Frobenius theorem, there is a unique positive invariant probability \({\varvec{\nu }}\) on \({\mathbb {X}}\) satisfying the following property: there exist \(c_1>0\) and \(c_2>0\) such that for any function \(g \in {\mathscr {C}}\) and \(n \geqslant 1\),
where \({\varvec{\nu }}(g) = \sum _{x\in {\mathbb {X}}} g(x) {\varvec{\nu }}(x)\).
The following two hypotheses ensure that the Markov walk has no drift and is non-lattice, respectively.
Hypothesis M2
The function f is centred:
Hypothesis M3
For any \((\theta ,a) \in {\mathbb {R}}^2\), there exists a sequence \(x_0, \dots , x_n\) in \({\mathbb {X}}\) such that
and
Under Hypothesis M1, it is shown in Sect. 4 that Hypothesis M3 is equivalent to the condition that the perturbed operator \({\mathbf {P}}_t \) has a spectral radius less than 1 for \( t\ne 0\); for more details we refer to Sect. 4. Furthermore, in the Appendix (see Lemma 10.3, Sect. 10), we show that Hypotheses M1–M3 imply that the following number \(\sigma ^2\), which is the limit of \({\mathbb {E}}_x ( S_n^2 )/n\) as \(n \rightarrow +\infty \) for any \(x \in {\mathbb {X}}\), is not zero:
Under spectral gap assumptions, the asymptotic behaviour of the probability \({\mathbb {P}}_x \left( \tau _y > n \right) \) and of the conditional law of the Markov walk \(\frac{y+S_n}{\sqrt{n}}\) given the event \(\{ \tau _y > n \}\) have been studied in [14]. It is easy to see that under M1, M2 and (2.2) the conditions of [14] are satisfied (see Sect. 10). We summarize the main results of [14] in the following propositions.
Proposition 2.1
(Preliminary results, part I) Assume Hypotheses M1–M3. There exists a non-degenerate non-negative function V on \({\mathbb {X}} \times {\mathbb {R}}\) such that
- 1.
For any \((x,y) \in {\mathbb {X}} \times {\mathbb {R}}\) and \(n \geqslant 1\),
$$\begin{aligned} {\mathbb {E}}_x \left( V \left( X_n, y+S_n \right) ;\, \tau _y > n \right) = V(x,y). \end{aligned}$$ - 2.
For any \(x \in {\mathbb {X}}\), the function \(V(x,\cdot )\) is non-decreasing and for any \((x,y) \in {\mathbb {X}} \times {\mathbb {R}}\),
$$\begin{aligned} V(x,y) \leqslant c \left( 1+\max (y,0) \right) . \end{aligned}$$ - 3.
For any \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\) and \(\delta \in (0,1)\),
$$\begin{aligned} \left( 1- \delta \right) \max (y,0) - c_{\delta } \leqslant V(x,y) \leqslant \left( 1+\delta \right) \max (y,0) + c_{\delta }. \end{aligned}$$
Since the function V satisfies the point 1, it is said to be harmonic for the killed Markov walk \((y+S_n)_{n\geqslant 0}\).
Proposition 2.2
(Preliminary results, part II) Assume Hypotheses M1–M3.
- 1.
For any \((x,y) \in {\mathbb {X}} \times {\mathbb {R}}\),
$$\begin{aligned} \lim _{n\rightarrow +\infty } \sqrt{n}{\mathbb {P}}_x \left( \tau _y > n \right) = \frac{2V(x,y)}{\sqrt{2\pi } \sigma }, \end{aligned}$$where \(\sigma \) is defined by (2.2).
- 2.
For any \((x,y) \in {\mathbb {X}} \times {\mathbb {R}}\) and \(n\geqslant 1\),
$$\begin{aligned} {\mathbb {P}}_x \left( \tau _y > n \right) \leqslant c\frac{ 1 + \max (y,0) }{\sqrt{n}}. \end{aligned}$$
Define the support of V by
Note that from property 3 of Proposition 2.1, for any fixed \(x\in {\mathbb {X}}\), the function \(y \mapsto V(x,y)\) is positive for large y. For further details on the properties of \(supp(V)\) we refer to [14].
Proposition 2.3
(Preliminary results, part III) Assume Hypotheses M1–M3.
- 1.
For any \((x,y) \in supp(V)\) and \(t\geqslant 0\),
where \({\varvec{\Phi }}^+(t) = 1-e^{-\frac{t^2}{2}}\) is the Rayleigh distribution function.
- 2.
There exists \(\varepsilon _0 >0\) such that, for any \(\varepsilon \in (0,\varepsilon _0)\), \(n\geqslant 1\), \(t_0 > 0\), \(t\in [0,t_0]\) and \((x,y) \in {\mathbb {X}} \times {\mathbb {R}}\),
$$\begin{aligned} \left| {\mathbb {P}}_x \left( y+S_n \leqslant t \sqrt{n} \sigma ,\, \tau _y > n \right) - \frac{2V(x,y)}{\sqrt{2\pi n}\sigma } {\varvec{\Phi }}^+(t) \right| \leqslant c_{\varepsilon ,t_0} \frac{\left( 1+\max (y,0)^2 \right) }{n^{1/2+\varepsilon }}. \end{aligned}$$
In the point 1 of Proposition 2.2 and the point 2 of Proposition 2.3, the function V can be zero, so that for all pairs (x, y) satisfying \(V(x,y)=0\) it holds
and
We note that, for the convenience of the reader, Propositions 2.1, 2.2 and 2.3 are formulated here under Hypotheses M1–M3, but they can be stated under much more general conditions, in particular for Markov chains with countable state spaces, see [14].
Now we proceed to formulate the main results of the paper. Our first result is an extension of Gnedenko–Stone local limit theorem originally stated for sums of independent random variables. The following theorem generalizes it to the case of sums of random variables defined on Markov chains conditioned to stay positive.
Theorem 2.4
Assume Hypotheses M1–M3. Let \(a>0\) be a positive real. Then there exists \(\varepsilon _0 \in (0,1/4)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\), non-negative function \(\psi \in {\mathscr {C}}\), \(y \in {\mathbb {R}}\) and \( n \geqslant 3\varepsilon ^{-3}\), we have
where \(\varphi _+(t) = te^{-\frac{t^2}{2}} \mathbb {1}_{\{t\geqslant 0\}}\) is the Rayleigh density and the constants c and \(c_{\varepsilon }\) may depend on a.
Note that Theorem 2.4 is meaningful only for large values of z such that \(z \sim n^{1/2} \) as \(n\rightarrow \infty \). Indeed, the remainder term is of order \(n^{-1-\varepsilon },\) with some small \(\varepsilon >0,\) while for a fixed z the leading term is of order \(n^{-3/2}\). When \(z = cn^{1/2}\) the leading term becomes of order \(n^{-1}\) while the remainder is still \(o(n^{-1})\). To deal with the case of z in compact sets a more refined result will be given below. We will deduce it from Theorem 2.4, however for the proof we need the concept of duality.
Let us introduce the dual Markov chain and the corresponding associated Markov walk. Since \({\varvec{\nu }}\) is positive on \({\mathbb {X}}\), the following dual Markov kernel \({\mathbf {P}}^*\) is well defined:
It is easy to see that \({\varvec{\nu }}\) is also \({\mathbf {P}}^*\)-invariant. The dual of \(( X_n)_{n\geqslant 0}\) is the Markov chain \(\left( X_n^* \right) _{n\geqslant 0}\) with values in \({\mathbb {X}}\) and transition probability \({\mathbf {P}}^*\). Without loss of generality we can consider that the dual Markov chain \(\left( X_n^* \right) _{n\geqslant 0}\) is defined on an extension of the probability space \((\varOmega , {\mathscr {F}}, {\mathbb {P}})\) and that it is independent of the Markov chain \(( X_n)_{n\geqslant 0}\). We define the associated dual Markov walk by
For any \(z\in {\mathbb {R}}\), define also the exit time
For any \(\in {\mathbb {X}}\), denote by \({\mathbb {P}}_x^*\) and \({\mathbb {E}}_x^*\) the probability, respectively the expectation, generated by the finite dimensional distributions of the Markov chain \(( X_n^* )_{n\geqslant 0}\) starting at \(X_0^* = x\). It is shown in Sect. 3 that the dual Markov chain \(\left( X_n^* \right) _{n\geqslant 0}\) satisfies Hypotheses M1–M3 as do the original chain \(\left( X_n \right) _{n\geqslant 0}\). Thus, Propositions 2.1–2.3 hold also for \(\left( X_n^* \right) _{n\geqslant 0}\) with V, \(\tau ,\)\((S_n)_{n\geqslant 0}\) and \({\mathbb {P}}_x\) replaced by \(V^*,\)\(\tau ^*,\)\((S_n^*)_{n\geqslant 0}\) and \({\mathbb {P}}_x^*\). Note also that both chains have the same invariant probability \({\varvec{\nu }}\). Denote by \({\mathbb {E}}_{{\varvec{\nu }}}\), \({\mathbb {E}}_{{\varvec{\nu }}}^*\) the expectations generated by the finite dimensional distributions of the Markov chains \(( X_n )_{n\geqslant 0}\) and \(( X_n^* )_{n\geqslant 0}\) in the stationary regime.
Our second result is a conditional version of the local limit theorem for fixed x, y and z.
Theorem 2.5
- 1.
For any non-negative function \(\psi \in {\mathscr {C}}\), \(a>0\), \(x\in {\mathbb {X}}\), \(y \in {\mathbb {R}}\) and \(z \geqslant 0\),
$$\begin{aligned} \lim _{n\rightarrow +\infty } n^{3/2}&{\mathbb {E}}_x \left( \psi \left( X_{n} \right) ;\, y+S_{n} \in [z,z+a],\, \tau _y> n \right) \\&\qquad = \frac{2V(x,y)}{\sqrt{2\pi }\sigma ^3} \int _z^{z+a} {\mathbb {E}}_{{\varvec{\nu }}}^* \left( \psi \left( X_1^* \right) V^*\left( X_1^*, z'+S_1^* \right) ;\, \tau _{z'}^* > 1 \right) \text {d}z'. \end{aligned}$$ - 2.
Moreover, there exists \(c > 0\) such that for any \(a>0\), non-negative function \(\psi \in {\mathscr {C}}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\) and \(n \geqslant 1\),
$$\begin{aligned}&\sup _{x\in {\mathbb {X}}} {\mathbb {E}}_x \left( \psi \left( X_{n} \right) ;\, y+S_{n} \in [z,z+a],\, \tau _y > n \right) \\&\quad \leqslant \frac{c \left\| \psi \right\| _{\infty }}{n^{3/2}} \left( 1+a^3 \right) \left( 1+z \right) \left( 1+\max (y,0) \right) . \end{aligned}$$
In the particular case when \(\psi =1\), the previous theorem rewrites as follows:
Corollary 2.6
- 1.
For any \(a>0\), \(x\in {\mathbb {X}}\), \(y \in {\mathbb {R}}\) and \(z \geqslant 0\),
$$\begin{aligned} \lim _{n\rightarrow +\infty } n^{3/2}&{\mathbb {P}}_x \left( y+S_{n} \in [z,z+a],\, \tau _y > n \right) \\&\qquad = \frac{2V(x,y)}{\sqrt{2\pi }\sigma ^3} \int _z^{z+a} \int _{{\mathbb {X}}} V^*\left( x', z' \right) {\varvec{\nu }} (\text {d}x') \text {d}z'. \end{aligned}$$ - 2.
Moreover, there exists \(c > 0\) such that for any \(a>0\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\) and \(n \geqslant 1\),
$$\begin{aligned} \sup _{x\in {\mathbb {X}}} {\mathbb {P}}_x \left( y+S_{n} \in [z,z+a],\, \tau _y > n \right) \leqslant \frac{c}{n^{3/2}} \left( 1+a^3 \right) \left( 1+z \right) \left( 1+\max (y,0) \right) . \end{aligned}$$
Note that the assertion 1 of Theorem 2.5 and assertion 1 of Corollary 2.6 hold for fixed \(a>0\), \(x\in {\mathbb {X}}\), \(y \in {\mathbb {R}}\) and \(z \geqslant 0\) and that these results do not cover the case when z is not in a compact set, for instance when \(z \sim n^{1/2}\).
The following result extends Theorem 2.5 to some functionals of the trajectories of the chain \(( X_n )_{n\geqslant 0}\). For any \((x,x^*) \in {\mathbb {X}}^2\), the probability generated by the finite dimensional distributions of the two dimensional Markov chain \(( X_n, X_n^*)_{n\geqslant 0}\) starting at \((X_0,X_0^*) = (x,x^*)\) is given by \({\mathbb {P}}_{x,x^*}={\mathbb {P}}_{x} \times {\mathbb {P}}_{x^*}^*\). Let \({\mathbb {E}}_{x,x^*}\) be the corresponding expectation. For any \(l \geqslant 1\), denote by \({\mathscr {C}}^+ ( {\mathbb {X}}^l \times {\mathbb {R}}_+ )\) the set of non-negative functions g: \({\mathbb {X}}^l \times {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+\) satisfying the following properties:
for any \((x_1,\dots ,x_l) \in {\mathbb {X}}^l\), the function \(z \mapsto g(x_1,\dots ,x_l,z)\) is continuous,
there exists \(\varepsilon > 0\) such that \(\max _{x_1,\dots x_l \in {\mathbb {X}}} \sup _{z \geqslant 0} g(x_1,\dots ,x_l,z) (1+z)^{2+\varepsilon } < +\infty \).
Theorem 2.7
Assume Hypotheses M1–M3. For any \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(l \geqslant 1\), \(m \geqslant 1\) and \(g \in {\mathscr {C}}^+ \left( {\mathbb {X}}^{l+m} \times {\mathbb {R}}_+ \right) \),
As a consequence of Theorem 2.7 we deduce the following asymptotic behaviour of the probability of the event \(\left\{ \tau _y=n \right\} \) as \(n\rightarrow +\infty \).
Theorem 2.8
Assume Hypotheses M1–M3. For any \(x\in {\mathbb {X}}\) and \(y \in {\mathbb {R}}\),
3 Properties of the dual Markov chain
In this section we establish some properties of the dual Markov chain and of the corresponding Markov walk.
Lemma 3.1
Suppose that the operator \({\mathbf {P}}\) satisfies Hypotheses M1–M3. Then the dual operator \({\mathbf {P}}^*\) satisfies also M1–M3.
Proof
By the definition of \({\mathbf {P}}^*,\) for any \(x^* \in {\mathbb {X}}\),
which proves that \({\varvec{\nu }}\) is also \({\mathbf {P}}^*\)-invariant. Thus Hypothesis M2, \({\varvec{\nu }}(f) = {\varvec{\nu }}(-f)=0\), is satisfied for both chains. Moreover, it is easy to see that for any \(n \geqslant 1\), \((x,x^*) \in {\mathbb {X}}^2\),
This shows that \({\mathbf {P}}^*\) satisfies M1 and M3. \(\square \)
Note that the operator \({\mathbf {P}}^*\) is the adjoint operator of \({\mathbf {P}}\) in the space \(L^2 \left( {\varvec{\nu }} \right) :\) for any functions g and h on \({\mathbb {X}},\)
In particular for any \(n\geqslant 1\), \({\varvec{\nu }} \left( f \left( \mathbf {P}^*\right) ^n f \right) = {\varvec{\nu }} \left( f {\mathbf {P}}^n f \right) \) and we note that
The following assertion plays a key role in the proofs.
Lemma 3.2
(Duality) For any probability measure \({\mathfrak {m}}\) on \({\mathbb {X}}\), any \(n\geqslant 1\) and any function F from \({\mathbb {X}}^n\) to \({\mathbb {R}}\),
Proof
We write
By the definition of \({\mathbf {P}}^*\), we have
and the result of the lemma follows. \(\square \)
4 The perturbed operator
For any \(t \in {\mathbb {R}}\), denote by \({\mathbf {P}}_t\) the perturbed transition operator defined by
where \({\mathbf {i}}\) is the complex \({\mathbf {i}}^2 = -1\). Let also \(r_t\) be the spectral radius of \({\mathbf {P}}_t\). Note that for any \(g \in {\mathscr {C}}\), \(\left\| {\mathbf {P}}_t g\right\| _{\infty } \leqslant \left\| e^{{\mathbf {i}}tf} g\right\| _{\infty } = \left\| g\right\| _{\infty }\) and so
We introduce the two following definitions:
A sequence \(x_0, x_1, \dots , x_n \in {\mathbb {X}}\), is a path (between \(x_0\) and \(x_n\)) if
$$\begin{aligned} {\mathbf {P}}(x_0,x_1) \cdots {\mathbf {P}}(x_{n-1},x_n) > 0. \end{aligned}$$A sequence \(x_0, x_1, \dots , x_n \in {\mathbb {X}}\), is an orbit if \(x_0, x_1, \dots , x_n, x_0\) is a path.
Note that under Hypothesis M1, for any \(x_0, x\in {\mathbb {X}}\) it is always possible to connect \(x_0\) and x by a path \(x_0, x_1, \dots , x_n, x\) in \({\mathbb {X}}\).
Lemma 4.1
Assume Hypothesis M1. The following statements are equivalent:
- 1.
There exists \((\theta ,a) \in {\mathbb {R}}^2\) such that for any orbit \(x_0, \dots , x_n\) in \({\mathbb {X}}\), we have
$$\begin{aligned} f(x_0) + \cdots + f(x_n) - (n+1)\theta \in a{\mathbb {Z}}. \end{aligned}$$ - 2.
There exist \(t\in {\mathbb {R}}^*\), \(h\in {\mathscr {C}}{\setminus } \{0\}\) and \(\theta \in {\mathbb {R}}\) such that for any \((x,x') \in {\mathbb {X}}^2\),
$$\begin{aligned} h(x')e^{{\mathbf {i}}tf(x')}{\mathbf {P}}(x,x') = h(x) e^{{\mathbf {i}}t\theta } {\mathbf {P}}(x,x'). \end{aligned}$$ - 3.
There exists \(t \in {\mathbb {R}}^*\) such that
$$\begin{aligned} r_t = 1. \end{aligned}$$
Proof
The point 1 implies the point 2. Suppose that the point 1 holds. Fix \(x_0 \in {\mathbb {X}}\) and set \(h(x_0) = 1\). For any \(x \in {\mathbb {X}}\), define h(x) in the following way: for any path \(x_0, \dots , x_n, x\) in \({\mathbb {X}}\) we set
where \(t = \frac{2\pi }{a}\). Note that if \(a=0\), then the point 1 holds also for \(a=1\) and so, without lost of generality, we assume that \(a\ne 0\). We first verify that h is well defined on \({\mathbb {X}}\). Recall that under Hypothesis M1, for any \(x\in {\mathbb {X}}\) it is always possible to connect \(x_0\) and x by a path. We have to check that the value of h(x) does not depend on the choice of the path. Let \(p,q \geqslant 1\) and \(x_0,x_1, \dots , x_p, x\) in \({\mathbb {X}}\) and \(x_0,y_1, \dots , y_q, x\) in \({\mathbb {X}}\) be two paths between \(x_0\) and x. We complete these paths to orbits as follows. Under Hypothesis M1, there exist \(n \geqslant 1\) and \(z_1, \dots , z_n\) in \({\mathbb {X}}\) such that
i.e. the sequence \(x, z_1, \dots , z_n, x_0\) is a path. So, the sequences \(x_0,x_1,\dots ,x_p,x,z_1,\dots ,z_n\) and \(x_0,y_1,\dots ,y_q,x,z_1,\dots ,z_n\) are orbits. By the point 1, there exist \(l_1,l_2 \in {\mathbb {Z}}\) such that
Therefore,
and since \(ta=2\pi \) it proves that h is well defined. Now let \((x,x') \in {\mathbb {X}}^2\) be such that \({\mathbf {P}}(x,x') > 0\). There exists a path \(x_0, x_1, \dots , x_n, x\) between \(x_0\) and x and so
Since \(x_0,x_1, \dots , x_n,x,x'\) is a path between \(x_0\) and \(x'\), we have also
Note that since the modulus of h is 1, this function belongs to \({\mathscr {C}} {\setminus }\{0\}\).
The point 2 implies the point 1 Suppose that the point 2 holds and let \(x_0, \dots , x_n\) be an orbit. Using the point 2 repeatedly, we have
Since h is a non-identically zero function with a constant modulus, necessarily, h is never equal to 0 and so \(f(x_0)+\cdots +f(x_n) -(n+1)\theta \in \frac{2\pi }{t} {\mathbb {Z}}\).
The point 2 implies the point 3 Suppose that the point 2 holds. Summing on \(x'\) we have, for any \(x \in {\mathbb {X}}\),
Therefore h is an eigenvector of \({\mathbf {P}}_t\) associated to the eigenvalue \(e^{{\mathbf {i}} t \theta }\) which implies that \(r_t \geqslant \left|e^{{\mathbf {i}} t \theta }\right| = 1\) and by (4.1), \(r_t = 1\).
The point 3 implies the point 2 Suppose that the point 3 holds. There exist \(h \in {\mathscr {C}} {\setminus } \{0\}\) and \(\theta \in {\mathbb {R}}\) such that \({\mathbf {P}}_t h = he^{{\mathbf {i}} t \theta }\). Without loss of generality, we suppose that \(\left\| h\right\| _{\infty } = 1\). Since \({\mathbf {P}}_t^n h = he^{{\mathbf {i}} t n \theta }\) for any \(n \geqslant 1\), by (2.1), for any \(x \in {\mathbb {X}}\), we have
From (4.2), letting \(x_0 \in {\mathbb {X}}\) be such that \(\left|h(x_0)\right| = \left\| h\right\| _{\infty } = 1\), it is easy to see that
From this it follows that the modulus of h is constant on \({\mathbb {X}}\): \(\left|h(x)\right| = \left|h(x_0)\right| = 1\) for any \(x \in {\mathbb {X}}\). Consequently, there exists \(\alpha \): \({\mathbb {X}} \rightarrow {\mathbb {R}}\) such that for any \(x \in {\mathbb {X}}\),
With (4.3) the equation \({\mathbf {P}}_t h = he^{{\mathbf {i}} t \theta }\) can be rewritten as
Since \(e^{{\mathbf {i}} \alpha (x)} e^{{\mathbf {i}} t \theta } \in \left\{ z \in {\mathbb {C}} : \left|z\right|=1 \right\} \) and \(e^{{\mathbf {i}} \alpha (x')} e^{{\mathbf {i}} f(x')} \in \left\{ z \in {\mathbb {C}} : \left|z\right|=1 \right\} \), for any \(x' \in {\mathbb {X}}\), the previous equation holds only if \(h(x')e^{{\mathbf {i}} t f(x')} = e^{{\mathbf {i}} \alpha (x')} e^{{\mathbf {i}} t f(x')} = e^{{\mathbf {i}} \alpha (x)} e^{{\mathbf {i}} t \theta } = h(x) e^{{\mathbf {i}} t \theta }\) for any \(x' \in {\mathbb {X}}\) such that \({\mathbf {P}}(x,x') > 0\). \(\square \)
Define the operator norm \(\left\| \cdot \right\| _{{\mathscr {C}} \rightarrow {\mathscr {C}}}\) on \({\mathscr {C}}\) as follows: for any operator R: \({\mathscr {C}} \rightarrow {\mathscr {C}}\), set
Lemma 4.2
Assume Hypotheses M1 and M3. For any compact set K included in \({\mathbb {R}}^*\) there exist constants \(c_K > 0\) and \(c_K' >0\) such that for any \(n \geqslant 1\),
Proof
By Lemma 4.1, under Hypotheses M1 and M3, we have \(r_t \ne 1\) for any \(t\ne 0\) and hence, using (4.1),
It is well known that
Since \(t \mapsto {\mathbf {P}}_t\) is continuous, the function \(t \mapsto r_t\) is the infimum of the sequence of upper semi-continuous functions \(t \mapsto \left\| {\mathbf {P}}_t^n\right\| _{{\mathscr {C}} \rightarrow {\mathscr {C}}}^{1/n}\) and therefore is itself upper semi-continuous. In particular, for any compact set K included in \({\mathbb {R}}^*\), there exists \(t_0 \in K\) such that
We deduce that for \(\varepsilon = (1- \sup _{t\in K} r_t)/2 >0\) there exists \(n_0 \geqslant 1\) such that for any \(n \geqslant n_0\),
Choosing \(c_{K'} = -\ln \left( \sup _{t\in K} r_t + \varepsilon \right) \) and \(c_K = \max _{n\leqslant n_0} \left\| {\mathbf {P}}_t^n\right\| _{{\mathscr {C}} \rightarrow {\mathscr {C}}} e^{c_{K'}n} +1\), the lemma is proved. \(\square \)
In the proofs we make use of the following assertion which is a consequence of the perturbation theory of linear operators (see for example [23]). The point 5 is proved in Lemma 2 of Guivarc’h and Hardy [18].
Proposition 4.3
Assume Hypotheses M1 and M2. There exist a real \(\varepsilon _0>0\) and operator valued functions \(\varPi _t\) and \(Q_t\) acting from \([-\varepsilon _0,\varepsilon _0]\) to the set of operators onto \({\mathscr {C}} \) such that
- 1.
the maps \(t \mapsto \varPi _t\), \(t \mapsto Q_t\) and \(t \mapsto \lambda _t\) are analytic at 0,
- 2.
the operator \({\mathbf {P}}_t\) has the following decomposition,
$$\begin{aligned} {\mathbf {P}}_t = \lambda _t \varPi _t +Q_t, \qquad \forall t \in [-\varepsilon _0,\varepsilon _0], \end{aligned}$$ - 3.
for any \(t\in [-\varepsilon _0,\varepsilon _0]\), \(\varPi _t\) is a one-dimensional projector and \(\varPi _t Q_t = Q_t \varPi _t = 0\),
- 4.
there exist \(c_1>0\) and \(c_2>0\) such that, for any \(n\in {\mathbb {N}}^*\),
$$\begin{aligned} \sup _{t\in [-\varepsilon _0,\varepsilon _0]} \left\| Q_t^n\right\| _{{\mathscr {C}} \rightarrow {\mathscr {C}}} \leqslant c_1 e^{-c_2 n}, \end{aligned}$$ - 5.
the function \(\lambda _t\) has the following expansion at 0: for any \(t \in [-\varepsilon _0,\varepsilon _0]\),
$$\begin{aligned} \left|\lambda _t - 1 + \frac{t^2 \sigma ^2}{2}\right| \leqslant c \left|t\right|^3. \end{aligned}$$
Note that \(\lambda _0=1\) and \(\varPi _0(\cdot ) = \varPi (\cdot ) = {\varvec{\nu }} (\cdot ) e\), where e is the unit function of \({\mathbb {X}}\): \(e(x) = 1\), for any \(x\in {\mathbb {X}}\).
Lemma 4.4
Assume Hypotheses M1 and M2. There exists \(\varepsilon _0 > 0\) such that for any \(n \geqslant 1\) and \(t \in [-\varepsilon _0 \sqrt{n}, \varepsilon _0\sqrt{n}]\),
Proof
By the points 2 and 3 of Proposition 4.3, for any \(t/\sqrt{n} \in [-\varepsilon _0,\varepsilon _0]\),
By the points 1 and 4 of Proposition 4.3, for \(n\geqslant 1\),
Let \(\alpha \) be the complex valued function defined on \([-\varepsilon _0,\varepsilon _0]\) by \(\alpha (t) = \frac{1}{t^3} \left( \lambda _t - 1 + \frac{t^2 \sigma ^2}{2} \right) \) for any \(t \in [-\varepsilon _0,\varepsilon _0] {\setminus } \{0\}\) and \(\alpha (0) = 0\). By the point 5 of Proposition 4.3, there exists \(c >0\) such that
With this notation, we have for any \(t/\sqrt{n} \in [-\varepsilon _0,\varepsilon _0]\),
Without loss of generality, the value of \(\varepsilon _0> 0\) can be chosen such that \(\varepsilon _0^2 \sigma ^2 \leqslant 1\) and so for any \(t/\sqrt{n} \in [-\varepsilon _0,\varepsilon _0]\), we have \(1-\frac{t^2 \sigma ^2}{2n} \geqslant 1/2\). Therefore,
Using the inequality \(1+u \leqslant e^{u}\) for \(u \in {\mathbb {R}}\), the fact that \(1-\frac{t^2 \sigma ^2}{2n} \geqslant 1/2\) and the bound (4.6), we have
Next, using the inequality \(e^{u}-1 \leqslant u e^{u}\) for \(u \geqslant 0\) and the fact that \(\left|t\right|/\sqrt{n} \leqslant \varepsilon _0\),
Again, without loss of generality, the value of \(\varepsilon _0> 0\) can be chosen such that \(c \varepsilon _0^2 \leqslant \sigma ^2/8\) (this have no impact on (4.6) which holds for any \([-\varepsilon _0',\varepsilon _0'] \subseteq [-\varepsilon _0,\varepsilon _0]\)). Thus, from (4.8) it follows that
Using the inequalities \(1-u \leqslant e^{-u}\) for \(u \in {\mathbb {R}}\) and \(\ln (1-u) \geqslant -u-u^2\) for \(u \leqslant 1\), we have
Putting together (4.7), (4.9) and (4.10), we obtain that, for any \(t/\sqrt{n} \in [-\varepsilon _0,\varepsilon _0]\),
In the same way, one can prove that
The right hand side in the assertion of the lemma can be bounded as follows:
Using (4.4), (4.5), (4.11) and (4.12), we obtain that, for any \(t/ \sqrt{n} \in [\varepsilon _0,\varepsilon _0]\),
\(\square \)
5 A non asymptotic local limit theorem
In this section we establish a local limit theorem for the Markov walk jointly with the Markov chain. Our result is similar to that in Grama and Le Page [15] where the case of sums of independent random variables is considered under the Cramér condition. We refer to Guivarc’h and Hardy [18] for a local limit theorem for Markov chains with compact state spaces. In contrast to previous results for Markov chains our local limit theorem gives an explicit dependence of the constant in the remainder term on the target function h applied to the random walk \(y+S_n\) (see Lemmata 5.1, 5.4 and Corollary 5.5). All these results are stated for Markov chains with finite state spaces to shorten the exposition, but, a closer analysis of the proofs shows that, under appropriate spectral gap assumptions, these assertions can be extended to more general Markov chains, including the chains with denumerable state spaces.
We first establish a local limit theorem for integrable functions with Fourier transforms with compact supports. For any integrable function h: \({\mathbb {R}} \rightarrow {\mathbb {R}}\) denote by \({\widehat{h}}\) its Fourier transform:
When \({\widehat{h}}\) is integrable, by the inversion formula,
For any integrable functions h and g, let
be the convolution of h and g. Denote by \(\varphi _{\sigma }\) the density of the centred normal law with variance \(\sigma ^2\):
Lemma 5.1
Assume Hypotheses M1–M3. For any \(A > 0\), any integrable function h on \({\mathbb {R}}\) whose Fourier transform \({\widehat{h}}\) has a compact support included in \([-A,A]\), any real function \(\psi \) defined on \({\mathbb {X}}\) and any \(n \geqslant 1\),
Proof
By the inversion formula and the Fubini theorem,
Since \({\widehat{h}}( t ) = 0\) for any \(t \notin [-A,A]\), we write
where \(\varepsilon _0\) is defined by Lemma 4.4.
Bound of\(I_1\) By Lemma 4.2, for any \(\varepsilon _0 \leqslant \left|t\right| \leqslant A\), we have
Consequently,
Bound of\(I_2\) Substituting \(s=t\sqrt{n}\), we write
By Lemma 4.4, for any \(\left|s\right|\leqslant \varepsilon _0 \sqrt{n}\), we have
Therefore,
Putting together (5.2), (5.3) and (5.4), concludes the proof. \(\square \)
We extend the result of Lemma 5.1 for any integrable function (with not necessarily integrable Fourier transform). As in Stone [32], we introduce the kernel \(\kappa \) defined on \({\mathbb {R}}\) by
The function \(\kappa \) is integrable and its Fourier transform is given by
Note that
For any \(\varepsilon >0\), we define the function \(\kappa _{\varepsilon }\) on \({\mathbb {R}}\) by
Its Fourier transform is given by \({\widehat{\kappa }}_{\varepsilon } (t) = {\widehat{\kappa }}(\varepsilon t)\). Note also that, for any \(\varepsilon > 0\), we have
For any non-negative and locally bounded function h defined on \({\mathbb {R}}\) and any \(\varepsilon >0\), let \({\overline{h}}_{\varepsilon }\) and \({\underline{h}}_{\varepsilon }\) be the “thickened” functions: for any \(u \in {\mathbb {R}}\),
For any \(\varepsilon > 0\), denote by \({\mathscr {H}}_{\varepsilon }\) the set of non-negative and locally bounded functions h such that h, \({\overline{h}}_{\varepsilon }\) and \({\underline{h}}_{\varepsilon }\) are measurable from \(\left( {\mathbb {R}}, {\mathscr {B}} \left( {\mathbb {R}} \right) \right) \) to \(\left( {\mathbb {R}}_+, {\mathscr {B}} \left( {\mathbb {R}}_+ \right) \right) \) and Lebesgue-integrable (where \({\mathscr {B}} \left( {\mathbb {R}} \right) \), \({\mathscr {B}} \left( {\mathbb {R}}_+ \right) \) are the Borel \(\sigma \)-algebras).
Lemma 5.2
For any function \(h \in {\mathscr {H}}_{\varepsilon }\), \(\varepsilon \in (0,1/4)\) and \(u \in {\mathbb {R}}\),
Proof
Note that for any \(\left|v\right| \leqslant \varepsilon \) and \(u\in {\mathbb {R}}\), we have \(u\in [ u- v - \varepsilon , u- v+\varepsilon ]\). So,
Using the fact that \(\int _{{\mathbb {R}}} \kappa _{\varepsilon ^2} (u) \text {d}u = 1\) and (5.5), we write
Therefore,
For any \(\varepsilon \in (0,1/4)\),
Moreover, from (5.6),
\(\square \)
Lemma 5.3
Let \(\varepsilon >0\) and \(h \in {\mathscr {H}}_{\varepsilon }\).
- 1.
For any \(y \in {\mathbb {R}}\) and \(n\geqslant 1\),
$$\begin{aligned} \sqrt{n} \left( {\overline{h}}_{\varepsilon }*\kappa _{\varepsilon ^2} \right) *\varphi _{\sqrt{n}\sigma }(y) \leqslant \sqrt{n}\left( h*\varphi _{\sqrt{n}\sigma } \right) (y) + c \left\| {\overline{h}}_{2\varepsilon } - h\right\| _{L^1} + c \varepsilon \left\| h\right\| _{L^1}, \end{aligned}$$where \(\varphi _{\sqrt{n}\sigma }(\cdot )\) is defined by (5.1).
- 2.
For any \(y \in {\mathbb {R}}\) and \(n\geqslant 1\),
$$\begin{aligned} \sqrt{n}\left( {\overline{h}}_{\varepsilon }*\kappa _{\varepsilon ^2} \right) *\varphi _{\sqrt{n}\sigma }(y) \leqslant c \left\| {\overline{h}}_{\varepsilon }\right\| _{L^1}. \end{aligned}$$ - 3.
For any \(y\in {\mathbb {R}}\) and \(n\geqslant 1\),
$$\begin{aligned} \sqrt{n} \left( {\underline{h}}_{\varepsilon }*\kappa _{\varepsilon ^2} \right) *\varphi _{\sqrt{n}\sigma }(y) \geqslant \sqrt{n} \left( h*\varphi _{\sqrt{n}\sigma }\right) (y) - c \left\| h-{\underline{h}}_{2\varepsilon }\right\| _{L^1} - c \varepsilon \left\| h\right\| _{L^1}. \end{aligned}$$
Proof
For any \(\varepsilon > 0\), \(\left|v\right| \leqslant \varepsilon \) and \(u\in {\mathbb {R}}\) it holds \([u-v-\varepsilon ,u-v+\varepsilon ] \subset [u-2\varepsilon ,u+2\varepsilon ]\). Therefore,
Consequently, for any \(u\in {\mathbb {R}}\),
From this, using the bound \(\sqrt{n}\varphi _{\sqrt{n}\sigma }(\cdot ) \leqslant 1/(\sqrt{2\pi }\sigma )\) and (5.5), we obtain that
Using again the bound \(\sqrt{n}\varphi _{\sqrt{n}\sigma }(\cdot ) \leqslant 1/(\sqrt{2\pi }\sigma )\), we get
which proves the claim 1.
In the same way,
which establishes the claim 2.
Integrating this inequality and using once again the bound \(\sqrt{n} \varphi _{\sqrt{n}\sigma }(\cdot ) \leqslant \frac{1}{\sqrt{2\pi }\sigma }\), we have
Inserting h, we conclude that
\(\square \)
We are now equipped to prove a non-asymptotic theorem for a large class of functions h.
Lemma 5.4
Assume Hypotheses M1–M3. Let \(\varepsilon \in (0,1/4)\). For any function \(h \in {\mathscr {H}}_{\varepsilon }\), any non-negative function \(\psi \in {\mathscr {C}}\) and any \(n \geqslant 1\),
where \(\varphi _{\sqrt{n}\sigma }(\cdot )\) is defined by (5.1). Moreover,
Proof
We prove upper and lower bounds for \(\sqrt{n}{\mathbb {E}}_x \left( h\left( y+S_n \right) \psi \left( X_n \right) \right) \) from which the claim will follow.
The upper bound By Lemma 5.2, we have, for any \(x\in {\mathbb {X}}\), \(n\geqslant 1\), \(y\in {\mathbb {R}}\) and \(\varepsilon \in (0,1/4)\),
Since \({\overline{h}}_{\varepsilon }\) is integrable, the function \(u\mapsto {\overline{h}}_{\varepsilon }*\kappa _{\varepsilon ^2}(u)\) is integrable and its Fourier transform \(u\mapsto \widehat{{\overline{h}}}_{\varepsilon }(u) {\widehat{\kappa }}_{\varepsilon ^2}(u)\) has a support included in \([-1/\varepsilon ^2,1/\varepsilon ^2]\). Consequently, by Lemma 5.1,
Using the points 1 and 2 of Lemma 5.3 and the fact that \(\left|{\varvec{\nu }} \left( \psi \right) \right| \leqslant \left\| \psi \right\| _{\infty }\), we deduce that
Note that \(\left\| {\overline{h}}_{\varepsilon }*\kappa _{\varepsilon ^2}\right\| _{L^1} = \left\| {\overline{h}}_{\varepsilon }\right\| _{L^1}\) and
Consequently,
From (5.8), taking into account that \(\sqrt{n} \left( h*\varphi _{\sqrt{n}\sigma } \right) (y) \leqslant c\left\| h\right\| _{L^1}\), we deduce, in addition, that
The lower bound By Lemma 5.2, we write that
Bound of\(I_1\) The Fourier transform of the convolution \({\underline{h}}_{\varepsilon }*\kappa _{\varepsilon ^2}\) has a compact support included in \([-1/\varepsilon ^2,1/\varepsilon ^2]\). So by Lemma 5.1,
Using the point 3 of Lemma 5.3 and the fact that \(\left|{\varvec{\nu }} \left( \psi \right) \right| \leqslant \left\| \psi \right\| _{\infty }\),
Since \(\left\| {\underline{h}}_{\varepsilon }*\kappa _{\varepsilon ^2}\right\| _{L^1} = \left\| {\underline{h}}_{\varepsilon }\right\| _{L^1} \leqslant \left\| h\right\| _{L^1}\) and since \(\left\| \widehat{{\underline{h}}_{\varepsilon }*\kappa _{\varepsilon ^2}}\right\| _{L^1} \leqslant \left\| {\underline{h}}_{\varepsilon }\right\| _{L^1} \left\| {\widehat{\kappa }}_{\varepsilon ^2}\right\| _{L^1} = \frac{1}{\varepsilon ^2}\left\| {\underline{h}}_{\varepsilon }\right\| _{L^1} \leqslant \frac{1}{\varepsilon ^2} \left\| h\right\| _{L^1}\), we deduce that
Bound of\(I_2\) With the notation \(g_{\varepsilon ,v}(u) = {\underline{h}}_{\varepsilon } \left( u - v \right) \), we have
Consequently, using (5.9), we find that
Note that, for any u and \(v \in {\mathbb {R}}\),
So, \(\left\| \overline{\left( g_{\varepsilon ,v} \right) }_{2\varepsilon }\right\| _{L^1} \leqslant \left\| {\overline{h}}_{2\varepsilon }\right\| _{L^1}\) and
By (5.5),
Putting together (5.10), (5.11) and (5.12), we obtain that
Putting together the upper bound (5.8) and the lower bound (5.13), the first inequality of the lemma follows. The second inequality is proved in (5.9). \(\square \)
We now apply Lemma 5.4 when the function h is an indicator of an interval.
Corollary 5.5
Assume Hypotheses M1–M3. For any \(a>0\), \(\varepsilon \in (0,1/4)\), any non-negative function \(\psi \in {\mathscr {C}}\) and any \(n\geqslant 1\),
where \(\varphi _{\sqrt{n}\sigma }(\cdot )\) is defined by (5.1). In particular, there exists \(c> 0\) such that for any \(a >0\),
Proof
Let \(z \geqslant 0\), \(a>0\), \(\varepsilon \in (0,1/4)\). For any \(y \in {\mathbb {R}}\) set
It is clear that
where by convention \(\mathbb {1}_{[z+\varepsilon ,z+a-\varepsilon ]}(y) = 0\) when \(a \leqslant 2\varepsilon \). It is also easy to see that
Taking into account these last equalities and using Lemma 5.4, we find that
Moreover, the convolution \(\mathbb {1}_{[z,z+a]}*\varphi _{\sqrt{n}\sigma }\) is equal to
where \(\varPhi _{\sqrt{n}\sigma }(t) = \int _{-\infty }^{t} \frac{e^{-\frac{u^2}{2n\sigma ^2}}}{\sqrt{2\pi n}\sigma } \text {d}u\) is the distribution function of the centred normal law of variance \(n\sigma ^2\). By the Taylor-Lagrange formula, there exists \(\xi \in (y-z-a,y-z)\) such that
Using the fact that \(\sup _{u\in {\mathbb {R}}} \left|u\right|e^{-u^2} \leqslant c\),
Putting together (5.15) and (5.16), we conclude that
\(\square \)
6 Auxiliary bounds
We state two bounds on the expectation \({\mathbb {E}}_x \left( \psi (X_n) \,;\, y+S_n \in [z,z+a],\, \tau _y > n \right) \). The first one is of order 1 / n and independent of z. Then we reverse the Markov chain to improve it to a bound of order \(1/n^{3/2}.\) We refer to Denisov and Wachtel [8] for related results in the case of lattice valued independent random variables.
Lemma 6.1
Assume Hypotheses M1–M3. There exists \(c > 0\) such that for any \(a >0\), non-negative function \(\psi \in \mathscr {C}\), \(y \in {\mathbb {R}}\) and \(n \geqslant 1\)
Proof
We split the time n into two parts \(k := \left\lfloor n/2\right\rfloor \) and \(n-k\). By the Markov property,
Using the uniform bound (5.14) in Corollary 5.5, we obtain that
By the point 2 of Proposition 2.2, we get
Since \(n-k \geqslant n/2\) and \(k \geqslant n/4\) for any \(n \geqslant 4\), the lemma is proved (the case when \(n\leqslant 4\) is trivial). \(\square \)
Lemma 6.2
Assume Hypotheses M1–M3. There exists \(c > 0\) such that for any \(a >0\), non-negative function \(\psi \in \mathscr {C}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\) and \(n \geqslant 1\)
Proof
Set again \(k=\left\lfloor n/2\right\rfloor \). By the Markov property
Using Lemma 3.2 with \({\mathfrak {m}} = {\varvec{\delta }}_{x'}\) and
we have
By the Markov property,
where
On the event \(\left\{ y'+f\left( X_k^* \right) + \cdots + f\left( X_1^* \right) \in \left[ z,z+a \right] \right\} = \left\{ z+a + S_k^* \in \left[ y',y'+a \right] \right\} \), we have
So, for any \(y' > 0\),
Using Lemma 6.1 we have uniformly in \(y' >0\),
Putting together (6.3) and (6.1) and using the point 2 of Proposition 2.2,
Since \(n-k \geqslant n/2\) and \(k \geqslant n/4\) for any \(n \geqslant 4\), the lemma is proved. \(\square \)
7 Proof of Theorem 2.4
The aim of this section is to bound
uniformly in the end point z. The point is to split the time n into \(n=n_1+n_2\), where \(n_2 = \left\lfloor \varepsilon ^3 n\right\rfloor \) and \(n_1 = n- \left\lfloor \varepsilon ^3 n\right\rfloor \), and \(\varepsilon \in (0,1)\). Using the Markov property, we shall bound the process between \(n_1\) and n by the local limit theorem (Corollary 5.5) and between 1 and \(n_1\) by the integral theorem (Proposition 2.3). Following this idea we write
For the ease of reading the bounds of \(E_1\) and \(E_2\) are given in separate sections.
7.1 Control of \(E_1\)
Lemma 7.1
Assume Hypotheses M1–M3. For any \(a>0\) and \(\varepsilon \in (0, 1/4)\) there exist \(c = c_a >0\) depending only on a and \(c_{\varepsilon }>0\) such that for any non-negative function \(\psi \in {\mathscr {C}}\), any \(y \in {\mathbb {R}}\) and \(n\ \in {\mathbb {N}}\), such that \(\varepsilon ^3 n\geqslant 1\) we have
where \(E_1 = {\mathbb {E}}_x \left( \psi \left( X_{n} \right) ;\, y+S_{n} \in [z,z+a],\, \tau _y > n_1 \right) \), \(n_2= \left\lfloor \varepsilon ^3 n\right\rfloor \), \(n_1 = n- \left\lfloor \varepsilon ^3 n\right\rfloor \) and \(\varphi (t) = e^{-\frac{t^2}{2}}/\sqrt{2\pi }\).
Proof
By the Markov property,
From now on we consider that the real \(a>0\) is fixed. By Corollary 5.5, for any \(\varepsilon ^{5/2} \leqslant \varepsilon \in (0,1/4)\),
with c depending only on a. Consequently, using (7.3) and the fact that \(n_2 = \left\lfloor \varepsilon ^3 n\right\rfloor \geqslant c_{\varepsilon } n\),
Therefore, by (5.1) and the point 2 of Proposition 2.2, we obtain that
Since \(n_2 \geqslant \varepsilon ^3 n \left( 1 - \frac{1}{\varepsilon ^3 n} \right) \) and \(n_1 \geqslant \frac{n}{2}\), we have
and the lemma follows. \(\square \)
To find the limit behaviour of \(E_1\), we will develop \(\frac{1}{\sqrt{n_2}}{\mathbb {E}}_x \left( \varphi \left( \frac{y+S_{n_1}-z}{\sqrt{n_2}\sigma } \right) ;\, \tau _y > n_1 \right) \). To this aim, we prove the following lemma which we will apply first with the standard normal density function \(\varphi \), and later on with the Rayleigh density \(\varphi _+\).
Lemma 7.2
Assume Hypotheses M1–M3. Let \(\varPsi \): \({\mathbb {R}} \rightarrow {\mathbb {R}}\) be a non-negative differentiable function such that \(\varPsi (t) \rightarrow 0\) as \(t \rightarrow +\infty \). Moreover we suppose that \(\varPsi '\) is a continuous function on \({\mathbb {R}}\) such that \(\max (\left|\varPsi (t)\right|,\left|\varPsi '(t)\right|) \leqslant c e^{-\frac{t^2}{4}}\). There exists \(\varepsilon _0 \in (0,1/2)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\), \(y \in {\mathbb {R}}\), \(m_1 \geqslant 1\) and \(m_2 \geqslant 1\), we have
where \(\varphi _+(t) = te^{-\frac{t^2}{2}}\).
Proof
Let \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\), \(m_1 \geqslant 1\) and \(m_2 \geqslant 1\) and fix \(\varepsilon _1 \in (0,1)\). We consider two cases. Assume first that \(z \leqslant \sqrt{m_1}\sigma /\varepsilon _1\). Using the regularity of the function \(\varPsi \), we note that
Denote by \(J_1\) the following integral:
Using the point 2 of Proposition 2.3, with \(t_0 = 2/\varepsilon _1\), there exists \(\varepsilon _0 > 0\) such that for any \(\varepsilon \in (0,\varepsilon _0)\),
By the point 2 of Proposition 2.1 and the point 2 of Proposition 2.2, with \(\left\| \varPsi '\right\| _{\infty } = \sup _{t\in {\mathbb {R}}} \left|\varPsi '(t)\right|\),
Since \(z \leqslant \frac{\sqrt{m_1}\sigma }{\varepsilon _1}\), we have \(\frac{2}{\varepsilon _1} - \frac{z}{\sqrt{m_1}\sigma } \geqslant \frac{1}{\varepsilon _1} \geqslant 1\) and so
Moreover, by the definition of \(J_1\) in (7.4), we have
Now, assume that \(z > \frac{\sqrt{m_1}\sigma }{\varepsilon _1}\). We write
Using the points 3 and 1 of Proposition 2.1, we can verify that
So by the point 2 of Proposition 2.2 and the point 2 of Proposition 2.1,
In the same way,
From the last two bounds it follows that for any \(z > \frac{\sqrt{m_1}\sigma }{\varepsilon _1}\),
Putting together (7.6), (7.7) and (7.5) and taking \(\varepsilon _1 = \varepsilon ^4\), we obtain the desired inequality for any \(z \geqslant 0\),
\(\square \)
Lemma 7.3
Assume Hypotheses M1–M3. There exists \(\varepsilon _0 \in (0,1/2)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\), \(y \in {\mathbb {R}}\), \(n \in {\mathbb {N}}\) such that \(\varepsilon ^3 n \geqslant 1\), we have
where \(\varphi (t) = e^{-\frac{t^2}{2}}/\sqrt{2\pi }\), \(\varphi _+(t) = te^{-\frac{t^2}{2}} \mathbb {1}_{\{t\geqslant 0\}}\), \(n_2 = \left\lfloor \varepsilon ^3 n\right\rfloor \) and \(n_1 = n-\left\lfloor \varepsilon ^3 n\right\rfloor \).
Proof
Denote
and
where \(\varphi _{\{\cdot \}}(\cdot )\) is defined in (5.1). By Lemma 7.2 we have
Since \(\frac{n}{2} \leqslant n_1 \leqslant n\) and \(\varepsilon ^3n-1 \leqslant n_2 \leqslant \varepsilon ^3 n\),
Let \(J_2\) be the following term:
Using (7.8),
By the point 2 of Proposition 2.1, we write
Putting together (7.9) and (7.11), we obtain that
It remains to link \(J_2\) from (7.10) to the desired equivalent. We distinguish two cases. If \(\frac{z}{\sigma } \leqslant \frac{\sqrt{n}}{\varepsilon }\),
If \(\frac{z}{\sigma } > \frac{\sqrt{n}}{\varepsilon } \geqslant \frac{\sqrt{n_1}}{\varepsilon }\), we have
Therefore, using the point 2 of Proposition 2.1, we obtain that in each case
Putting together (7.12) and (7.13), proves the lemma. \(\square \)
Another consequence of Lemma 7.2 is the following lemma which will be used in Sect. 8.
Lemma 7.4
Assume Hypotheses M1–M3. There exists \(\varepsilon _0 \in (0,1/2)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\), \(y \in {\mathbb {R}}\), \(n \in {\mathbb {N}}\) such that \(\varepsilon ^3 n \geqslant 2\), we have
where \(\varphi _+(t) = te^{-\frac{t^2}{2}} \mathbb {1}_{\{t\geqslant 0\}}\) is the Rayleigh density function, \(n_1 = n-\left\lfloor \varepsilon ^3n\right\rfloor \) and \(n_2 = \left\lfloor \varepsilon ^3n\right\rfloor \).
Proof
Using Lemma 7.2 with \(\varPsi = \varphi _+\), \(m_1=n_1\), \(m_2 = n_2-1\) and \(z=0,\)
where
and
where \(\varphi _{\{\cdot \}}(\cdot )\) is defined in (5.1). So,
By the point 2 of Proposition 2.1,
The lemma follows from (7.14) and (7.15). \(\square \)
Thanks to Lemmata 7.1 and 7.3 we can bound \(E_1\) from (7.2) as follows.
Lemma 7.5
Assume Hypotheses M1–M3. For any \(a > 0\) there exists \(\varepsilon _0 \in (0, 1/4)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\), any non-negative function \(\psi \in {\mathscr {C}}\), any \(y \in {\mathbb {R}}\) and \(n \in {\mathbb {N}}\) such that \(\varepsilon ^3 n \geqslant 1\), we have
where \(E_1 = {\mathbb {E}}_x \left( \psi \left( X_{n} \right) ;\, y+S_{n} \in [z,z+a],\, \tau _y > n_1 \right) \), \(n_1 = n- \left\lfloor \varepsilon ^3 n\right\rfloor \) and \(\varphi _+\) is the Rayleigh density function: \(\varphi _+(t) = te^{-\frac{t^2}{2}} \mathbb {1}_{\{t\geqslant 0\}}\).
Proof
From Lemmas 7.1 and 7.3, it follows that
\(\square \)
7.2 Control of \(E_2\)
In this section we bound the term \(E_2\) defined by (7.2). To this aim let us recall and introduce some notations: for any \(\varepsilon \in (0,1)\), we consider \(n_2 = \left\lfloor \varepsilon ^3 n\right\rfloor \), \(n_1 = n-n_2 = n-\left\lfloor \varepsilon ^3 n\right\rfloor \), \(n_3 = \left\lfloor \frac{n_2}{2}\right\rfloor \) and \(n_4 = n_2-n_3\). We define also
and we note that
Lemma 7.6
Assume Hypotheses M1–M3. For any \(a>0\) there exists \(\varepsilon _0 \in (0,1/4)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\), any non-negative function \(\psi \in {\mathscr {C}}\), any \(y \in {\mathbb {R}}\) and \(n \in {\mathbb {N}}\) such that \(\varepsilon ^3 n \geqslant 1\), we have
where \(E_{21}\) is given as in (7.16) by
and \(n_1 = n-\left\lfloor \varepsilon ^3 n\right\rfloor \).
Proof
Using the Markov property and the uniform bound (5.14) of Corollary 5.5, with \(n_2 = \left\lfloor \varepsilon ^3 n\right\rfloor \),
We note that \(\frac{\varepsilon \sqrt{n}}{\sigma \sqrt{n_1}} \leqslant \frac{\varepsilon }{\sigma \sqrt{1-\varepsilon ^3}} \leqslant \frac{2}{\sigma } \varepsilon \) and so by the point 2 of Proposition 2.3 with \(t_0=2\varepsilon /\sigma \):
Using the point 2 of Proposition 2.1 and taking into account that \(n_2 \geqslant \varepsilon ^3 n \left( 1-\frac{c_{\varepsilon }}{n} \right) \), \(n_1 \geqslant n/2\) and that \({\varvec{\Phi }}^+(t) \leqslant {\varvec{\Phi }}^+(t_0) \leqslant \frac{t_0^2}{2}\) for any \(t\in (0,t_0)\),
which implies the assertion of the lemma. \(\square \)
Lemma 7.7
Assume Hypotheses M1–M3. For any \(a>0\) there exists \(\varepsilon _0 \in (0,1/4)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\), any non-negative function \(\psi \in {\mathscr {C}}\), any \(y \in {\mathbb {R}}\), and \(n \in {\mathbb {N}}\) satisfying \(\varepsilon ^3 n \geqslant 2\), we have
where \(E_{22}\) is given as in (7.17) by
and \(n_1 = n-\left\lfloor \varepsilon ^3 n\right\rfloor \), \(n_2 = \left\lfloor \varepsilon ^3 n\right\rfloor \) and \(n_3 = \left\lfloor \frac{n_2}{2}\right\rfloor \).
Proof
By the Markov property,
Bound of\(E_{22}'\) By the Markov property and the uniform bound (5.14) in Corollary 5.5, with \(n_4 = n_2 - n_3 = n - n_1-n_3\),
Let \((B_t)_{t\geqslant 0}\) be the Brownian motion defined by Proposition 10.4. Denote by \(A_n\) the following event:
and by \({\overline{A}}_n\) its complement. We have
Note that for any \(x' \in {\mathbb {X}}\) and any \(y' > \varepsilon \sqrt{n}\),
where, for any \(y'' > 0\), \(\tau _{y''}^{bm}\) is the exit time of the Brownian motion starting at \(y''\) defined by (10.7). Since \(y' > \varepsilon \sqrt{n}\), it implies that
Since \(\sqrt{n}/\sqrt{n_3} \geqslant \sqrt{2}/\varepsilon ^{3/2}\),
Therefore, putting together (7.21) and (7.22) and using Proposition 10.4,
Since \(n_4 \geqslant n_2/2 \geqslant \frac{\varepsilon ^3 n}{2} \left( 1-\frac{c_{\varepsilon }}{n} \right) \) and \(n_3 \geqslant n_2/2-1 \geqslant \frac{\varepsilon ^3 n}{2} \left( 1-\frac{c_{\varepsilon }}{n} \right) \), we have
Inserting (7.23) in (7.20) and using the point 2 of Proposition 2.2 and the fact that \(n_1 \geqslant n/2\), we conclude that
\(\square \)
Lemma 7.8
Assume Hypotheses M1–M3. For any \(a>0\) there exists \(\varepsilon _0 \in (0,1/4)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\), any non-negative function \(\psi \in {\mathscr {C}}\), any \(y \in {\mathbb {R}}\), and \(n \in {\mathbb {N}}\) such that \(\varepsilon ^3 n \geqslant 3\), we have
where \(E_{23}\) is given as in (7.18) by
and \(n_1 = n-\left\lfloor \varepsilon ^3 n\right\rfloor \), \(n_2 = \left\lfloor \varepsilon ^3 n\right\rfloor \) and \(n_3 = \left\lfloor \frac{n_2}{2}\right\rfloor \).
Proof
By the Markov property,
We consider two cases: when \(z \leqslant \frac{\varepsilon \sqrt{n}}{2}\) and when \(z > \frac{\varepsilon \sqrt{n}}{2}\).
Fix first \(0 \leqslant z \leqslant \frac{\varepsilon \sqrt{n}}{2}\). Using Corollary 5.5, we have for any \(y' > \varepsilon \sqrt{n}\),
So, when \(0 \leqslant z \leqslant \frac{\varepsilon \sqrt{n}}{2}\), we have
Now we consider that \(z > \frac{\varepsilon \sqrt{n}}{2}\). Using Lemma 3.2 with \({\mathfrak {m}} = {\varvec{\delta }}_{x'}\) and
we obtain
By the Markov property,
where \(\psi _{x'}^*\) is a function defined on \({\mathbb {X}}\) by the equation (6.2). We note that, on the event \(\left\{ y'+f\left( X_{n_2}^* \right) + \cdots + f\left( X_1^* \right) \in \left[ z,z+a \right] \right\} = \left\{ z + S_{n_2}^* \in \left[ y'-a,y' \right] \right\} \), we have
Consequently,
with \(n_4 = n_2-n_3 = \left\lfloor \varepsilon ^3 n\right\rfloor - \left\lfloor \frac{\varepsilon ^3 n}{2}\right\rfloor \geqslant \frac{\varepsilon ^3n}{2} \left( 1-\frac{c_{\varepsilon }}{n} \right) \). Proceeding in the same way as for the term \(E_{22}'\) in (7.23) and using the fact that z is larger than \(c\varepsilon \sqrt{n}\), we have
Putting together (7.25) and (7.26), for any \(z \geqslant 0\), we obtain
Inserting this bound in (7.24) and using the point 2 of Proposition 2.2, we conclude that
\(\square \)
Putting together Lemmas 7.6, 7.7 and 7.8, by (7.19), we obtain the following bound for \(E_2\):
Lemma 7.9
Assume Hypotheses M1–M3. For any \(a>0\) there exists \(\varepsilon _0 \in (0,1/4)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\), any non-negative function \(\psi \in {\mathscr {C}}\), any \(y \in {\mathbb {R}}\) and \(n \in {\mathbb {N}}\) such that \(\varepsilon ^3 n \geqslant 3\), we have
where \(E_2\) is given as in (7.2) by
and \(n_1 = n-\left\lfloor \varepsilon ^3 n\right\rfloor \).
7.3 Proof of Theorem 2.4
Lemma 7.5 estimates \(E_1\) and Lemma 7.9 bounds \(E_2\). Taking into account these two lemmas, Theorem 2.4 follows.
8 Proof of Theorem 2.5
8.1 Preliminary results
Lemma 8.1
Assume Hypotheses M1–M3. For any \(a>0\) and \(p \in {\mathbb {N}}^*\), there exists \(\varepsilon _0 \in (0,1/4)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\) there exists \(n_0(\varepsilon ) \geqslant 1\) such that any non-negative function \(\psi \in {\mathscr {C}}\), any \(y' >0\), \(z \geqslant 0\), \(k\in \{0, \dots , p-1\}\) and \(n \geqslant n_0(\varepsilon )\), we have
and
where \(E_k' = {\mathbb {E}}_{x'} \left( \psi \left( X_{n_2} \right) ;\, y'+S_{n_2} \in \left( z_k,z_k+\frac{a}{p} \right] ,\, \tau _{y'} > n_2 \right) \), \(z_k =z+\frac{ka}{p}\) and \(n_2 = \left\lfloor \varepsilon ^3 n\right\rfloor \).
Proof
Using Lemma 3.2 with \({\mathfrak {m}} = {\varvec{\delta }}_{x'}\) and
we have
where \(\psi _{x'}^*\) is the function defined on \({\mathbb {X}}\) by (6.2).
The upper bound Note that, on the event \(\left\{ y'+f\left( X_{n_2}^* \right) + \cdots + f\left( X_1^* \right) \in \right. \left. \left( z_k,z_k+\frac{a}{p} \right] \right\} = \left\{ z_k +\frac{a}{p} + S_{n_2}^* \in \left[ y',y'+\frac{a}{p} \right) \right\} \), we have
So, for any \(y'>0\),
Using Theorem 2.4 for the reversed chain with \(\varepsilon '=\varepsilon ^{8}\), we obtain that
Note that by (6.2), \({\varvec{\nu }} \left( \psi _{x'}^* \right) = 1\) and \(\left\| \psi _{x'}^*\right\| _{\infty } \leqslant c\). So,
and the upper bound of the lemma is proved.
The lower bound Similarly as in the proof of the upper bound we note that, on the event \(\left\{ y'+f\left( X_{n_2}^* \right) + \cdots + f\left( X_1^* \right) \in \left( z_k,z_k+\frac{a}{p} \right] \right\} = \left\{ z_k + S_{n_2}^* \in \left[ y'-\frac{a}{p},y' \right) \right\} \), we have
Let \(y_+' := \max (y'-a/p,0)\) and \(a' := \min (y',a/p) \in (0,a]\). For any \(\eta \in (0,a')\),
Using Theorem 2.4,
Note that, if \(y' \geqslant a/p\) we have
and if \(0 < y' \leqslant a/p\) we have
Moreover, using the points 1 and 2 of Proposition 2.1, we observe that
Consequently, for any \(y' > 0\),
Taking the limit as \(\eta \rightarrow 0\), the lower bound of the lemma follows. \(\square \)
Lemma 8.2
Assume Hypotheses M1–M3. For any \(a>0\) and \(p \in {\mathbb {N}}^*\), there exists \(\varepsilon _0 \in (0,1/4)\) such that for any \(\varepsilon \in (0,\varepsilon _0)\) there exists \(n_0(\varepsilon ) \geqslant 1\) such that any non-negative function \(\psi \in {\mathscr {C}}\), any \(y \in {\mathbb {R}}\), \(z \geqslant 0\) and \(n \geqslant n_0(\varepsilon )\), we have
and
where \(E_0 = {\mathbb {E}}_{x} \left( \psi \left( X_n \right) ;\, y+S_n \in \left( z,z+a \right] ,\, \tau _y > n \right) \) and for any \(k\in \{0,\dots ,p-1\}\), \(z_k = z+\frac{ka}{p}\).
Proof
Set \(n_1 = n-\left\lfloor \varepsilon ^3 n\right\rfloor \) and \(n_2 = \left\lfloor \varepsilon ^3n\right\rfloor \). By the Markov property, for any \(p \geqslant 1\),
where for any \(k \in \{ 0, \dots , p-1 \}\),
and \(z_k = z+\frac{ka}{p}\).
The upper bound By Lemma 8.1,
where \(J_1(k)= {\mathbb {E}}_{{\varvec{\nu }}}^* \left( \psi \left( X_1^* \right) V^*\left( X_1^*, z_k+\frac{a}{p}+S_1^* \right) ;\, \tau _{z_k+\frac{a}{p}}^* > 1 \right) \), for any \(k\in \{0,\dots ,p-1\}\). By Lemma 7.4 and the point 2 of Proposition 2.2,
Note that, using the points 1 and 2 of Proposition 2.1, we have
Therefore
and the upper bound of the lemma is proved.
The lower bound The proof of the lower bound is similar to the proof of the upper bound and therefore will not be detailed. \(\square \)
8.2 Proof of Theorem 2.5
The second point of Theorem 2.5 was proved by Lemma 6.2. It remains to prove the first point. Let \(\psi \in {\mathscr {C}}\), \(a>0\), \(x\in {\mathbb {X}}\), \(y \in {\mathbb {R}}\) and \(z \geqslant 0\). Suppose first that \(z>0\). For any \(n \geqslant 1\) and \(\eta \in (0,\min (z,1))\),
where \(E_0(\eta ) = {\mathbb {E}}_{x} \left( \psi \left( X_n \right) ;\, y+S_n \in \left( z-\eta ,z+a \right] ,\, \tau _y > n \right) \). Taking the limit as \(n\rightarrow +\infty \) in Lemma 8.2, we have, for any \(p \in {\mathbb {N}}^*\) and \(\varepsilon \in (0, \varepsilon _0(p))\),
with \(z_{k,\eta } = z-\eta +\frac{k(a+\eta )}{p}\) for \(k\in \{0,\dots ,p-1\}\). Taking the limit as \(\varepsilon \rightarrow 0\),
By the point 2 of Proposition 2.1, the function \(u \mapsto V^*\left( x^*, u-f(x^*) \right) \mathbb {1}_{\left\{ u-f(x^*) >0 \right\} }\) is monotonic and so is Riemann integrable. Since \({\mathbb {X}}\) is finite, we have
Therefore,
Taking the limit as \(\eta \rightarrow 0\) and using (8.3), we obtain that, for any \(z >0\),
If \(z=0\), we have
Using Lemma 8.2 and the same arguments as before, it is easy to see that (8.4) holds for \(z=0\).
Since \([z,z+a] \supset (z,z+a]\) we have obviously
Using this and Lemma 8.2 we obtain (8.4) with \(\liminf \) instead of \(\limsup ,\) which concludes the proof of the theorem.
9 Proof of Theorems 2.7 and 2.8
9.1 Preliminaries results
Lemma 9.1
Assume Hypotheses M1–M3. For any \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\), \(a >0\), any non-negative function \(\psi \): \({\mathbb {X}} \rightarrow {\mathbb {R}}_+\) and any non-negative and continuous function g: \([z,z+a] \rightarrow {\mathbb {R}}_+\), we have
Proof
Fix \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\), \(a >0\), and let \(\psi \): \({\mathbb {X}} \rightarrow {\mathbb {R}}_+\) be a non-negative function and g: \([z,z+a] \rightarrow {\mathbb {R}}_+\) be a non-negative and continuous function. For any measurable non-negative and bounded function \(\varphi \): \({\mathbb {R}} \rightarrow {\mathbb {R}}_+\), we define
We first prove that for any \(0 \leqslant \alpha < \beta \) we have
Since \([\alpha ,\beta ) \subset [\alpha ,\beta ]\), the upper limit is a straightforward consequence of Theorem 2.5:
and for the lower limit, we write for any \(\eta \in (0,\beta -\alpha )\),
Taking the limit as \(\eta \rightarrow 0\), it proves (9.1).
From (9.1), by linearity, for any non-negative staircase function \(\varphi = \sum _{k=1}^N \gamma _k \mathbb {1}_{[\alpha _k,\beta _k)}\), where \(N \geqslant 1\), \(\gamma _1, \dots , \gamma _N \in {\mathbb {R}}_+\) and \(0< \alpha _1< \beta _1 = \alpha _2< \cdots < \beta _N\), we have
Since g is continuous on \([z,z+a]\), for any \(\varepsilon \in (0,1)\) there exists \(\varphi _{1,\varepsilon }\) and \(\varphi _{2,\varepsilon }\) two stepwise functions on \([z,z+a)\) such that \(g-\varepsilon \leqslant \varphi _{1,\varepsilon } \leqslant g \leqslant \varphi _{2,\varepsilon } \leqslant g+\varepsilon \). Consequently,
Taking the limit as \(\varepsilon \rightarrow 0\), concludes the proof of the lemma. \(\square \)
For any \(l \geqslant 1\) we denote by \({\mathscr {C}}_b^+ \left( {\mathbb {X}}^l \times {\mathbb {R}} \right) \) the set of measurable non-negative functions g: \({\mathbb {X}}^l \times {\mathbb {R}} \rightarrow {\mathbb {R}}_+\) bounded and such that for any \((x_1,\dots ,x_l) \in {\mathbb {X}}^l\), the function \(z \mapsto g(x_1,\dots ,x_l,z)\) is continuous.
Lemma 9.2
Assume Hypotheses M1–M3. For any \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\), \(a >0\), \(l \geqslant 1\), any non-negative functions \(\psi \): \({\mathbb {X}} \rightarrow {\mathbb {R}}_+\) and \(g\in \mathscr {C}_b^+ \left( {\mathbb {X}}^l \times {\mathbb {R}} \right) \), we have
Proof
We reduce the proof to the previous case using the Markov property. Fix \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\), \(a >0\), \(l \geqslant 1\), \(\psi \): \({\mathbb {X}} \rightarrow {\mathbb {R}}_+\) and \(g \in {\mathscr {C}}_b^+ \left( {\mathbb {X}}^l \times {\mathbb {R}} \right) \). For any \(n \geqslant l+1\), by the Markov property,
where for any \((x_1,\dots ,x_l) \in {\mathbb {X}}^l\), \(y' \in {\mathbb {R}}\) and \(k \geqslant 1\),
By the point 2 of Theorem 2.5,
Consequently, by the Lebesgue dominated convergence theorem (in fact the expectation \({\mathbb {E}}_x\) is a finite sum) and Lemma 9.1,
\(\square \)
Lemma 9.2 can be reformulated for the dual Markov walk as follows:
Lemma 9.3
Assume Hypotheses M1–M3. For any \(x' \in {\mathbb {X}}\), \(z \geqslant 0\), \(y' \geqslant 0\), \(a >0\), \(m \geqslant 1\) and any function \(g \in {\mathscr {C}}_b^+ \left( {\mathbb {X}}^m \times {\mathbb {R}} \right) \), we have
Proof
Fix \(x' \in {\mathbb {X}}\), \(z \geqslant 0\), \(y' \geqslant 0\), \(a >0\), \(m \geqslant 1\) and \(g \in {\mathscr {C}}_b^+ \left( {\mathbb {X}}^m \times {\mathbb {R}} \right) \). Let \(\psi _{x'}^*\) be the function defined on \({\mathbb {X}}\) by (6.2) and consider for any \(n \geqslant m+1\),
By Lemma 9.2 applied to the dual Markov walk, we have
Moreover, using (6.2) and the fact that \({\varvec{\nu }}\) is \({\mathbf {P}}\)-invariant, for any \(x' \in {\mathbb {X}}\), \(y'' \geqslant 0\),
By the point 1 of Proposition 2.1, the function V is harmonic and so
\(\square \)
Lemma 9.4
Assume Hypotheses M1–M3. For any \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\), \(a >0\), \(m \geqslant 1\) and any function \(g \in {\mathscr {C}}_b^+ \left( {\mathbb {X}}^m \times {\mathbb {R}} \right) \), we have
Proof
Fix \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\), \(a >0\), \(m \geqslant 1\) and \(g \in {\mathscr {C}}_b^+ \left( {\mathbb {X}}^m \times {\mathbb {R}} \right) \). For any \(n \geqslant m\), consider
For any \(l \geqslant 1\) and \(n \geqslant l+m\), by the Markov property, we have
For any \(p \geqslant 1\) and \(0 \leqslant k \leqslant p\) we define \(z_k := z+\frac{a k}{p}\). For any \(x' \in {\mathbb {X}}\), \(y' > 0\), \(n \geqslant l+m\) and \(p \geqslant 1\), we write
Using Lemma 3.2, we get
where \(\psi _{x'}^*\) is defined by (6.2).
The upper bound Using (8.1), we have
By Lemma 9.3,
where for any \(k \geqslant 0\) and \(t \in {\mathbb {R}}\),
Note that for any \(t \in [-a/p,0]\)
Since \(y'' \mapsto V\left( x', y'' \right) \) is non-decreasing (see the point 2 of Proposition 2.1), we have
Moreover, by (9.2) and the point 2 of Theorem 2.5,
Consequently, by (9.3) and the Lebesgue dominated convergence theorem (or using just the fact that \({\mathbb {X}}\) is finite),
Using the point 3 of Proposition 2.1, for any \(\delta \in (0,1)\),
and again using the point 3 of Proposition 2.1, for any \(\delta \in (0,1)\),
Using the point 1 of Proposition 2.1 and the point 2 of Proposition 2.2 and taking the limit as \(l \rightarrow +\infty \),
Taking the limit as \(\delta \rightarrow 0\),
For any \((x_1^*,\dots ,x_m^*) \in {\mathbb {X}}^m\) and \(u \in {\mathbb {R}}\), let
The function \(u \mapsto g_m(u)\) is uniformly continuous on \([z,z+a]\). Consequently, for any \(\varepsilon > 0\), there exists \(p_0 \geqslant 1\) such that for any \(p \geqslant p_0\),
Moreover, using the point 2 of Proposition 2.1, it is easy to see that the function \(u \mapsto V_m^*(u)\) is non-decreasing and so is Riemann-integrable. Therefore, as \(p \rightarrow +\infty \), we have
Thus, when \(\varepsilon \rightarrow 0\),
Moreover, since \(u \mapsto V_m^*(u)\) is non-decreasing,
Consequently, by the Lebesgue dominated convergence theorem, (9.4), (9.7) and the Fubini theorem,
By (9.5), we obtain that,
The lower bound Repeating similar arguments as in the upper bound, by (8.2), we have for any \(x' \in {\mathbb {X}}\), \(y' > 0\), \(l \geqslant 1\), \(n \geqslant l+m+1\), \(p \geqslant 1\),
where \(y_+' = \max (y'-a/p,0)\) and \(a' = \min (y',a/p) \in (0,a/p)\). Using Lemma 9.3,
where, for any \(t \in {\mathbb {R}}\),
Since \(y'' \mapsto V\left( x', y'' \right) \) is non-decreasing (see the point 2 of Proposition 2.1), we have
where
Moreover, by the point 3 of Proposition 2.1, for any \(\delta \in (0,1)\),
Consequently, using (9.3) and the Fatou Lemma,
Using the point 1 of Proposition 2.1 and the point 2 of Proposition 2.2 and taking the limit as \(l \rightarrow +\infty \) and then as \(\delta \rightarrow 0\),
Using the notation from (9.6) and the fact that \(u \mapsto g_m(u)\) is uniformly continuous on \([z,z+a]\), for any \(\varepsilon > 0\),
Taking the limit as \(\varepsilon \rightarrow 0\),
By the Fatou lemma, (9.8) and (9.9), we conclude that
\(\square \)
From now on, we consider that the dual Markov chain \(\left( X_n^* \right) _{n\geqslant 0}\) is independent of \(\left( X_n \right) _{n\geqslant 0}\). Recall that its transition probability \({\mathbf {P}}^*\) is defined by (2.4) and that, for any \(z \geqslant 0\), the associated Markov walk \(( z+S_n^* )_{n\geqslant 0}\) and the associated exit time \(\tau _z^*\) are defined by (2.5) and (2.6) respectively. Recall also that for any \((x,x^*) \in {\mathbb {X}}^2\), we denote by \({\mathbb {P}}_{x,x^*}\) and \({\mathbb {E}}_{x,x^*}\) the probability and the expectation generated by the finite dimensional distributions of the Markov chains \(( X_n )_{n\geqslant 0}\) and \(( X_n^* )_{n\geqslant 0}\) starting at \(X_0 = x\) and \(X_0^* = x^*\) respectively.
Lemma 9.5
Assume Hypotheses M1–M3. For any \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\), \(a >0\), \(l \geqslant 1\), \(m \geqslant 1\) and any function \(g \in {\mathscr {C}}_b^+ \left( {\mathbb {X}}^{l+m} \times {\mathbb {R}} \right) \), we have
Proof
Fix \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(z \geqslant 0\), \(a >0\), \(l \geqslant 1\), \(m \geqslant 1\) and \(g \in {\mathscr {C}}_b^+ \left( {\mathbb {X}}^{l+m} \times {\mathbb {R}} \right) \). For any \(n \geqslant l+m\), by the Markov property,
where \(y_l = x_1+\cdots +x_l\). Using the Lebesgue dominated convergence theorem (or simply the fact that \({\mathbb {X}}^l\) is finite) and Lemma 9.4, we conclude that
\(\square \)
9.2 Proof of Theorem 2.7
For any \(l \geqslant 1\), denote by \({\mathscr {C}}^+ ( {\mathbb {X}}^l \times {\mathbb {R}}_+ )\) the set of non-negative functions g: \({\mathbb {X}}^l \times {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+\) satisfying the following properties:
for any \((x_1,\dots ,x_l) \in {\mathbb {X}}^l\), the function \(z \mapsto g(x_1,\dots ,x_l,z)\) is continuous,
there exists \(\varepsilon > 0\) such that \(\max _{x_1,\dots x_l \in {\mathbb {X}}} \sup _{z \geqslant 0} g(x_1,\dots ,x_l,z) (1+z)^{2+\varepsilon } < +\infty \).
Fix \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\), \(l \geqslant 1\), \(m \geqslant 1\) and \(g \in {\mathscr {C}}^+ \left( {\mathbb {X}}^{l+m} \times {\mathbb {R}} \right) \). For brevity, denote
Set
Since \(g \in {\mathscr {C}}^+ \left( {\mathbb {X}}^{l+m} \times {\mathbb {R}} \right) \), we have
where \(N(g) = \max _{x_1,\dots ,x_{l+m} \in {\mathbb {X}}} \sup _{z \geqslant 0} g(x_1,\dots ,x_{l+m},z)(1+z)^{2+\varepsilon } <+\infty \). By the point 2 of Theorem 2.5, we have
Consequently, by the Lebesgue dominated convergence theorem,
By Lemma 9.5,
which establishes Theorem 2.7.
9.3 Proof of Theorem 2.8
Theorem 2.8 will be deduced from Theorem 2.7.
Let \(x \in {\mathbb {X}}\), \(y \in {\mathbb {R}}\) and \(n \geqslant 1\). Since \({\mathbb {X}}\) is finite we note that \(\left\| f\right\| _{\infty } = \sup _{x \in {\mathbb {X}}} \left|f(x)\right|\) exists. This implies
By the Markov property,
where, for any \((x',y') \in {\mathbb {X}} \times {\mathbb {R}}\),
Since \(g(x',\cdot )\) is a staircase function, for any \(\varepsilon > 0\) there exist two functions \(\varphi _{\varepsilon }\) and \(\psi _{\varepsilon }\) on \({\mathbb {X}} \times {\mathbb {R}}\) and \(N \subset {\mathbb {X}} \times {\mathbb {R}}\) such that
for any \(x' \in {\mathbb {X}}\), the functions \(\varphi _{\varepsilon }(x',\cdot )\) and \(\psi _{\varepsilon }(x',\cdot )\) are continuous and have a compact support included in \(\left[ -1,\left\| f\right\| _{\infty }+1 \right] \),
for any \((x',y') \in \left( {\mathbb {X}} \times {\mathbb {R}} \right) {\setminus } N\), it holds \(\varphi _{\varepsilon }(x',y') = g(x',y') = \psi _{\varepsilon }(x',y')\),
for any \((x',y') \in {\mathbb {X}} \times {\mathbb {R}}\), it holds \(0 \leqslant \varphi _{\varepsilon }(x',y') \leqslant g(x',y') \leqslant \psi _{\varepsilon }(x',y') \leqslant 1\),
the set N is sufficiently small:
$$\begin{aligned} \int _{-1}^{\left\| f\right\| _{\infty }+1} {\mathbb {E}}_{{\varvec{\nu }}}^* \left( V^*\left( X_1, z+S_1^* \right) ;\, \tau _z^* > 1,\, \left( X_1, z \right) \in N \right) \text {d}z \leqslant \varepsilon . \end{aligned}$$(9.10)
The upper bound For any \(\varepsilon > 0\), using Theorem 2.7, we have
Using the point 1 of Proposition 2.1,
Since \({\varvec{\nu }}\) is \({\mathbf {P}}^*\)-invariant, we have
Using the point 1 of Proposition 2.1,
Moreover, by (9.10), we get
Putting together (9.11), (9.12) and (9.13) and taking the limit as \(\varepsilon \rightarrow 0\), we obtain that
Lower bound In a similar way, using Theorem 2.7, we write
Using (9.12) and (9.13) and taking the limit as \(\varepsilon \rightarrow 0\), we obtain that
which together with (9.14) concludes the proof.
References
Bertoin, J., Doney, R.A.: On conditioning a random walk to stay nonnegative. Ann. Probab. 22(4), 2152–2167 (1994)
Bolthausen, E.: On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4(3), 480–485 (1976)
Borovkov, A.A.: On the asymptotic behavior of distributions of first-passage times, I. Math. Notes 75(1–2), 23–37 (2004)
Borovkov, A.A.: On the asymptotic behavior of distributions of first-passage times, II. Math. Notes 75(3–4), 322–330 (2004)
Caravenna, F.: A local limit theorem for random walks conditioned to stay positive. Probab. Theory Relat. Fields 133(4), 508–530 (2005)
Denisov, D., Wachtel, V.: Conditional limit theorems for ordered random walks. Electron. J. Probab. 15, 292–322 (2010)
Denisov, D., Wachtel, V.: Exit times for integrated random walks. Ann. Inst. Henri Poincaré Probab. Stat. 51(1), 167–193 (2015)
Denisov, D., Wachtel, V.: Random walks in cones. Ann. Probab. 43(3), 992–1044 (2015)
Doeblin, W., Fortet, R.: Sur les chaìnes à liaisons complètes. Bull. Soc. Math. France 65, 132–148 (1937)
Doney, R.A.: On the asymptotic behaviour of first passage times for transient random walk. Probab. Theory Relat. Fields 81(2), 239–246 (1989)
Eichelsbacher, P., König, W.: Ordered random walks. Electron. J. Probab. 13, 1307–1336 (2008)
Gnedenko, B.V.: On a local limit theorem of the theory of probability. Uspekhi Mat. Nauk 3(3), 187–194 (1948)
Grama, I., Lauvergnat, R., Le Page, É.: Limit theorems for affine Markov walks conditioned to stay positive. Ann. Inst. Henri Poincaré Probabilités et Statistiques 54(1), 529–568 (2018)
Grama, I., Lauvergnat, R., Le Page, É.: Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption. Ann. Probab. 46(4), 1807–1877 (2018)
Grama, I., Le Page, É.: Bounds in the local limit theorem for a random walk conditioned to stay positive. In: Modern Problems of Stochastic Analysis and Statistics. Springer Proceedings in Mathematics & Statistics, pp. 103–130. Springer (2017)
Grama, I., Le Page, É., Peigné, M.: On the rate of convergence in the weak invariance principle for dependent random variables with application to Markov chains. Colloq. Math. 134(1), 1–55 (2014)
Grama, I., Le Page, É., Peigné, M.: Conditioned limit theorems for products of random matrices. Probab. Theory Relat. Fields 168(3–4), 601–639 (2017)
Guivarc’h, Y., Hardy, J.: Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. IHP Probab. Stat. 24, 73–98 (1988)
Hennion, H., Hervé.: Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics, vol. 1766. Springer Berlin Heidelberg New York (2001)
Iglehart, D.L.: Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 2(4), 608–619 (1974)
Iglehart, D.L.: Random walks with negative drift conditioned to stay positive. J. Appl. Probab. 11(4), 742–751 (1974)
Ionescu Tulcea, C.T., Marinescu, G.: Théorie ergodique pour des classes d’opérations non complètement continues. Ann. Math. 52(1), 140–147 (1950)
Kato, T.: Perturbation Theory for Linear Operators. Springer Berlin Heidelberg, Berlin (1976)
Kolmogorov, A.N.: A local limit theorem for classical Markov chains. Izv. Akad. Nauk SSSR, Ser. Math. 13(4), 281–300 (1949)
Le Page, E.: Théorèmes limites pour les produits de matrices aléatoires. In: Probability Measures on Groups, pp. 258–303. Springer, Berlin, Heidelberg (1982). https://doi.org/10.1007/BFb0093229
Le Page, É., Peigné, M.: A local limit theorem on the semi-direct product of \({\mathbb{R}}^{*+}\) and \({\mathbb{R}}^d\). Ann. Inst. Henri Poincare (B) Probab. Stat. 33, 223–252 (1997)
Nagaev, S.V.: Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2(4), 378–406 (1954)
Nagaev, S.V.: More exact statement of limit theorems for homogeneous Markov chains. Theory Probab. Appl. 6(1), 62–81 (1961)
Presman, E.: Boundary problems for sums of lattice random variables, defined on a finite regular Markov chain. Theory Probab. Appl. 12(2), 323–328 (1967)
Presman, E.: Methods of factorization and a boundary problems for sums of random variables defined on a Markov chain. Izv. Akad. Nauk SSSR 33, 861–990 (1969)
Spitzer, F.: Principles of Random Walk. The University Series in Higher Mathematics. D. Van Nostrand, Princeton (1964)
Stone, C.: A local limit theorem for nonlattice multi-dimensional distribution functions. Ann. Math. Stat. 36(2), 546–551 (1965)
Varopoulos, NTh: Potential theory in conical domains. Math. Proc. Camb. Philos. Soc. 125(2), 335–384 (1999)
Varopoulos, NTh: Potential theory in conical domains. II. Math. Proc. Camb. Philos. Soc. 129(2), 301–320 (2000)
Vatutin, V.A., Wachtel, V.: Local probabilities for random walks conditioned to stay positive. Probab. Theory Relat. Fields 143(1–2), 177–217 (2008)
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Appendix
Appendix
1.1 The non degeneracy of the Markov walk
In [14], it is proved that the statements of Propositions 2.1–2.3 hold under more general assumptions (see Hypotheses M1-M5 of [14]). We will link these assumptions to our Hypotheses M1–M3. The assumptions M1-M3 in [14], with the Banach space \({\mathscr {C}}\), are well known consequences of Hypothesis M1 of this paper. Hypothesis M4 in [14] is also obvious with \(N=N_1 = \cdots = 0\). By Hypothesis M2, to obtain Hypothesis M5 of [14], it remains only to prove that \(\sigma \) defined by (2.2) is strictly positive. First we give a necessary and sufficient condition. Recall that the words path and orbit are defined in Sect. 4.
Lemma 10.1
Assume Hypothesis M1. The following statements are equivalent:
- 1.
The Cesáro mean of f on the orbits is constant: there exists \(m \in {\mathbb {R}}\) such that for any orbit \(x_0,\dots ,x_n\) we have
$$\begin{aligned} f(x_0) + \cdots + f(x_n) = (n+1)m. \end{aligned}$$ - 2.
There exist a constant \(m \in {\mathbb {R}}\) and a function \(h \in {\mathscr {C}}\) such that for any \((x,x') \in {\mathbb {X}}^2\),
$$\begin{aligned} {\mathbf {P}}(x,x') f(x') = {\mathbf {P}}(x,x') \left( h(x)-h(x')+m \right) . \end{aligned}$$ - 3.
The following real \({{\tilde{\sigma }}}^2\) is equal to 0
$$\begin{aligned} {{\tilde{\sigma }}}^2 = {\varvec{\nu }} \left( f^2 \right) - {\varvec{\nu }} \left( f \right) ^2 + 2 \sum _{n=1}^{+\infty } \left[ {\varvec{\nu }} \left( f {\mathbf {P}}^n f \right) - {\varvec{\nu }} \left( f \right) ^2 \right] = 0. \end{aligned}$$
Proof
The point 1 implies the point 2 Suppose that the point 1 holds. Fix \(x_0 \in {\mathbb {X}}\) and set \(h(x_0)= 0\). For any \(x \in {\mathbb {X}}\), we define h(x) in the following way: for any path \(x_0,x_1,\dots ,x_n,x\) in \({\mathbb {X}}\), we set
We shall verify that h is well defined. By Hypothesis M1, we can find at least a path to define h(x). Now we have to check that this definition does not depend on the choice of the path. Let \(x_0,x_1,\dots ,x_p,x\) and \(x_0,y_1,\dots ,y_q,x\) be two paths. By Hypothesis M1, there exists a path \(x,z_1, \dots , z_n,x_0\) in \({\mathbb {X}}\) between x and \(x_0\). Since \(x_0,x_1,\dots ,x_p,x,z_1,\dots ,z_n\) and \(x_0,y_1,\dots ,y_p,x,z_1,\dots ,z_n\) are two orbits, by the point 1, we have
and so the function h is well defined on \({\mathbb {X}}\). Now let \((x,x') \in {\mathbb {X}}^2\) such that \({\mathbf {P}}(x,x') > 0\). By Hypothesis M1, there exists \(x_0,x_1, \dots , x_n,x\) a path between \(x_0\) and x. Since
by the definition of h, we have
In particular
The point 2 implies the point 1 Suppose that the point 2 holds and let \(x_0,\dots ,x_n\) be an orbit. Using the point 2,
and the point 1 follows.
The point 2 implies the point 3 Suppose that the point 2 holds. Denote by \({{\tilde{f}}}\) the \({\varvec{\nu }}\)-centred function:
By the point 2, for any \(x\in {\mathbb {X}}\),
Using the fact that \({\varvec{\nu }}\) is \({\mathbf {P}}\)-invariant, we obtain that \({\varvec{\nu }} \left( {{\tilde{f}}} \right) = 0 = m-{\varvec{\nu }}(f)\) and so,
Consequently, by (10.2), \({\mathbf {P}}^n {{\tilde{f}}} = {\mathbf {P}}^{n-1} h - {\mathbf {P}}^n h\) for any \(n \geqslant 1\) and therefore,
Let
be the solution of the Poisson equation \({{\tilde{\varTheta }}} - {\mathbf {P}} {{\tilde{\varTheta }}} = {{\tilde{f}}}\), which by (2.1), is well defined. Taking the limit as \(n \rightarrow +\infty \) in (10.4) and using (2.1),
Therefore, for any \((x,x') \in {\mathbb {X}}^2\),
Using the point 2 and (10.3), it follows that
for any \((x,x') \in {\mathbb {X}}^2\) such that \({\mathbf {P}}(x,x') > 0\). Moreover,
Since \({\varvec{\nu }}\) is \({\mathbf {P}}\)-invariant,
By (10.5), we conclude that \({{\tilde{\sigma }}}^2=0\).
The point 3 implies the point 2 Suppose that the point 3 holds. By (10.6), for any \((x,x') \in {\mathbb {X}}\) such that \({\mathbf {P}}(x,x')>0\) we have
Let \(h = {\mathbf {P}} {{\tilde{\varTheta }}}\). Since \({{\tilde{\varTheta }}}\) is the solution of the Poisson equation,
By the definition of \({{\tilde{f}}}\) in (10.1), for any \((x,x') \in {\mathbb {X}}\) such that \({\mathbf {P}}(x,x')>0\),
with \(m = {\varvec{\nu }}(f)\). \(\square \)
Note that under Hypothesis M2, Lemma 10.1 can be rewritten as follows.
Lemma 10.2
Assume Hypotheses M1 and M2. The following statements are equivalent:
- 1.
The mean of f on the orbits is equal to zero: for any orbit \(x_0,\dots ,x_n\), we have
$$\begin{aligned} f(x_0) + \cdots + f(x_n) = 0. \end{aligned}$$ - 2.
There exists a function \(h \in {\mathscr {C}}\) such that for any \((x,x') \in {\mathbb {X}}^2\),
$$\begin{aligned} {\mathbf {P}}(x,x') f(x') = {\mathbf {P}}(x,x') \left( h(x)-h(x') \right) . \end{aligned}$$ - 3.
The real \(\sigma ^2\) is equal to 0:
$$\begin{aligned} \sigma ^2 = {\varvec{\nu }} \left( f^2 \right) + 2 \sum _{n=1}^{+\infty } {\varvec{\nu }} \left( f {\mathbf {P}}^n f \right) = 0. \end{aligned}$$
Now we prove that the Hypothesis M3 (the “non-lattice” condition), implies that the Markov walk has non-zero asymptotic variance.
Lemma 10.3
Under Hypotheses M1–M3, we have
Proof
We proceed by reductio ad absurdum. Suppose that \(\sigma ^2 = 0\). By Lemma 10.2, for any orbit \(x_0,\dots ,x_n\), we have
which implies the negation of Hypothesis M3 with \(\theta = a = 0\). \(\square \)
1.2 Strong approximation
Let \((B_t)_{t\geqslant 0}\) be the standard Brownian motion on \({\mathbb {R}}\) defined on the probability space \((\varOmega , {\mathscr {F}}, {\mathbb {P}})\). Consider the exit time
where \(\sigma \) is defined by (2.2). It is proved in Grama, Le Page and Peigné [16] that there is a version of the Markov walk \((S_n)_{n\geqslant 0}\) and of the standard Brownian motion \((B_t)_{t\geqslant 0}\) living on the same probability space which are close enough in the following sense:
Proposition 10.4
There exists \(\varepsilon _0 >0\) such that, for any \(\varepsilon \in (0,\varepsilon _0]\), \(x\in {\mathbb {X}}\) and \(n\geqslant 1\), without loss of generality (on an extension of the initial probability space) one can reconstruct the sequence \((S_n)_{n\geqslant 0}\) with a continuous time Brownian motion\((B_t)_{t\in {\mathbb {R}}_{+} }\), such that
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Grama, I., Lauvergnat, R. & Le Page, É. Conditioned local limit theorems for random walks defined on finite Markov chains. Probab. Theory Relat. Fields 176, 669–735 (2020). https://doi.org/10.1007/s00440-019-00948-8
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DOI: https://doi.org/10.1007/s00440-019-00948-8