Preliminaries of Lévy Processes

Abstract

Let $(\Omega , \fancyscript{F}, \mathrm P )$ be a probability space and $(R^d, \fancyscript{B}(R^d), \langle \cdot ,\cdot \rangle )$ be a d-dimensional Euclidean space $R^d$ with the $\sigma$-algebra of Borel subsets $\fancyscript{B}(R^d)$, the scalar product $\langle x,y\rangle =\sum\nolimits ^{d}_{j=1}x_jy_j$ for row vectors $x=(x_1,\ldots ,x_d)$, $y=(y_1,\ldots ,y_d)$ and the norm $|x|=\sqrt{\langle x,x\rangle }$.

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3.1 Lévy-Itô Decomposition

Let $(\Omega , \fancyscript{F}, \mathrm P )$ be a probability space and $(R^d, \fancyscript{B}(R^d), \langle \cdot ,\cdot \rangle )$ be a d-dimensional Euclidean space $R^d$ with the $\sigma$-algebra of Borel subsets $\fancyscript{B}(R^d)$, the scalar product $\langle x,y\rangle =\sum\nolimits ^{d}_{j=1}x_jy_j$ for row vectors $x=(x_1,\ldots ,x_d)$, $y=(y_1,\ldots ,y_d)$, and the norm $|x|=\sqrt{\langle x,x\rangle }$.

We are assuming that the reader is familiar with the foundations of probability theory based on the measure theory.

A mapping $X:R_+\times \Omega \rightarrow R^d$ such that for each $B\in \fancyscript{B}(R^d)$ and $t\ge 0$ $\left\{ \omega :X(t,\omega )\in B\right\} \in \fancyscript{F}$ is called a d-dimensional stochastic process.

For fixed $\omega \in \Omega$ a function $X(\cdot ,\omega )$ is called a sample path of $X$. Later we shall use the notation $X=\left\{ X_t,t\ge 0\right\}$.

If for each $t\ge 0$ and $\varepsilon >0$

$$\begin{aligned} \lim \limits _{h\rightarrow 0}\mathrm P \left\{ |X_{t+h}-X_t|>\varepsilon \right\} =0, \end{aligned}$$

a process $X=\{X_t,t\ge 0\}$ is called stochastically continuous.

Definition 3.1

A d-dimensional stochastic process $X=\left\{ X_t,t\ge 0\right\}$ is an additive process if the following conditions are satisfied:

• (1) for any $n\ge 1$ and $0\le t_0<t_1<\cdots <t_n$, increments $X_{t_0}, X_{t_1}-X_{t_0}, \ldots , X_{t_n}-X_{t_{n-1}}$ are independent;

• (2) $X_0=0$$\mathrm P$-a.e.;

• (3) $X$ is stochastically continuous;

• (4) $\mathrm P$-a.e. sample paths are right-continuous in $t\ge 0$ and have left limits in $t>0$.

An additive process in law is a stochastic process satisfying (1)–(3).

Let $X=\left\{ X_t, t\ge 0\right\}$ be a d-dimensional additive process.

Let $U_{\varepsilon }=\left\{ x\in R^d:|x|>\varepsilon \right\}$, $\varepsilon >0$, $\fancyscript{B}_{\varepsilon }(R^d)=\fancyscript{B}(R^d)\cap U_{\varepsilon }$.

For $B\in \fancyscript{B}_\varepsilon (R^d)$ define

$$\begin{aligned} p(t, B)=\sum \limits _{0\le s\le t}1_{B}(X_s-X_{s-}) \end{aligned}$$

and

$$\begin{aligned} X^{B}_t=\sum \limits _{0\le s\le t}(X_s-X_{s^-})1_B(X_s-X_{s-}), \quad t\ge 0. \end{aligned}$$

The following properties hold true (see, e.g., [1–3]).

• (i) For each $B\in \fancyscript{B}_{\varepsilon }(R^d)$, $t>0$, the function

  $$\begin{aligned} \mathrm E p(t,B):=\Pi (t,B)<\infty . \end{aligned}$$

  and is continuous in $t$. The stochastic process $p(t,B)$, $t\ge 0$ is a Poisson additive process with mean function $\Pi (t,B)$, $t\ge 0$, i. e. it satisfies the assumptions (1)–(4) and for each $t>0$, $k=0,1,\ldots$

  $$\begin{aligned} \mathrm P \left\{ p(t,B)=k\right\} =e^{-\Pi (t,B)}\frac{(\Pi (t,B))^k}{k!}. \end{aligned}$$

  Moreover,

  $$\begin{aligned} \int \limits _{R^d\backslash \{0\}}|x|^2\wedge 1\Pi (t,\text{ d}x)<\infty . \end{aligned}$$

• (ii) For each $B_1,\ldots ,B_m\in \fancyscript{B}_{\varepsilon }(R^d)$ such that $B_j\cap B_k=\emptyset$, $j\ne k$, stochastic processes

  $$\begin{aligned} \left\{ X_t^{B_1},t\ge 0\right\} ,\ldots ,\left\{ X_t^{B_m}, t\ge 0\right\} \quad \mathrm and \quad \left\{ X_t-\sum ^m_{j=1}X^{B_j}_j,t\ge 0\right\} \end{aligned}$$

  are additive mutually independent processes and for each $\varepsilon >0$, $t>0$

  $$\begin{aligned} \mathrm E |X_t-X_t^{U_{\varepsilon }}|^2<\infty . \end{aligned}$$

• (iii) Let $0<\varepsilon _n\downarrow 0$, as $n\rightarrow \infty$, and $\Delta _k=\left\{ x\in R^d:\varepsilon _k<|x|\le \varepsilon _{k-1}\right\}$, $k=2,3,\ldots$, $\Delta _1=\left\{ x\in R^d:|x|>\varepsilon _1\right\}$. There exists a subsequence $\left\{ n_k,k=1,2,\ldots \right\}$ such that, as $k\rightarrow \infty$, the sequence

  $$\begin{aligned} X_t^{(k)}:=X_t-X_t^{\Delta _1}-\sum \limits ^{n_k}_{j=2}(X_t^{\Delta _j}-EX_t^{\Delta _j}), \quad t\ge 0, \end{aligned}$$

  converges uniformly on each finite time interval P-a.e. to the continuous Gaussian additive process $X^0=\left\{ X_t^0,t\ge 0\right\}$ such that

  $$\begin{aligned} \mathrm E e^{i\langle z,X_t^0\rangle }=\exp \left\{ i\langle z,a(t)\rangle -\frac{1}{2}\langle zA(t),z\rangle \right\} , \quad z\in R^d, \end{aligned}$$

  where $a(t)$, $t\ge 0$, is a continuous d-dimensional function and $A(t)$ is a continuous symmetric nonnegative definite $d\times d$ matrix valued function.

• (iv) For each $z\in R^d$ and $t>0$

  $$\begin{aligned} \mathrm E \exp \left\{ i\langle z,X_t\rangle \right\}&=\exp \Bigg \{i\langle z,a(t)\rangle -\frac{1}{2}\langle zA(t),z\rangle \Bigg . \nonumber \\&\quad + \Bigg .\int \limits _{R^d\backslash \{0\}}\left(e^{i\langle z,x\rangle }-1-i\langle z,x\rangle 1_{\{|x|\le 1\}}\right)\Pi (t,\text{ d}x)\Bigg \}, \end{aligned}$$

  (3.1)

  implying the Lévy-Khinchine formula as $t=1$.

Definition 3.2

A d-dimensional additive process $X=\left\{ X_t,t\ge 0\right\}$ is called a Lévy process if it is temporally homogeneous, i.e., for each $s,t>0$,

$$\begin{aligned} \fancyscript{L}\left(X_{t+s}-X_s\right)=\fancyscript{L}(X_t). \end{aligned}$$

Definition 3.3

A d-dimensional stochastic process $X=\{X_t,t\ge 0\}$ is called a Lévy process in law if it is temporally homogeneous and satisfies the assumptions (1)–(3).

An additive process is a Lévy one if and only if the functions $a(t)$, $A(t)$ and $\Pi (t,B)$, $t\ge 0$, are linear in $t$, i.e., $a(t)=at$, $A(t)=At$ and $\Pi (t, B)=\Pi (B)t$. The triplet $(a, A, \Pi )$, where $a\in R^d$, $A$ is $a$ symmetric nonnegative definite $d\times d$ matrix and $\Pi (B)$, $B\in \fancyscript{B}(R^d_0)$, is a measure such that

$$\begin{aligned} \int \limits _{R^d_0}|x|^2\wedge 1 \Pi (\text{ d}x)<\infty , \end{aligned}$$

is called the triplet of Lévy characteristics; $R^d_0:=R^d\backslash \{0\}$. This triplet uniquely defines the finite dimensional distributions $\fancyscript{L}\left(X_{t_1},X_{t_2},\ldots ,X_{t_n}\right)$, $0\le t_1<t_2<\cdots <t_n$, $n\ge 1$.

The class of Lévy triplets corresponds one-to-one with the class of Lévy processes in law.

A d-dimensional Lévy process $X$ with the triplet of Lévy characteristics $(0,I_d,0)$, where $I_d$ is the $d\times d$ unit matrix, is called the standard d-dimensional Brownian motion .

Definition 3.4

A probability distribution $\mu$ on $R^d$ is called infinitely divisible if, for any positive integer $n$, there exists a probability measure $\mu _n$ on $R^d$ such that $\mu =\underbrace{\mu _{n}*\cdots *\mu _{n}}_{n \quad \mathrm times }$. Here “$*$” means the convolution of probability distributions.

We shall write that $\mu \in ID(R^d)$.

From the celebrated Lévy-Khinchine formula and (3.1) it follows that the class of infinitely distributions $\mu$ corresponds one-to-one with the class of Lévy processes in law by means of the equality

$$\begin{aligned}&\mathrm E \exp \left\{ i\langle z,X_1\rangle \right\} \\&\qquad =\exp \left\{ i\langle z,a\rangle -\frac{1}{2}\langle zA, z\rangle +\int \limits _{R_0^d}\left(e^{i\langle z, x\rangle }-1-i\langle z,x\rangle 1_{\left\{ |x|\le 1\right\} }\right)\Pi (\text{ d}x)\right\} \\&\qquad = \int \limits _{R^d}e^{i\langle z,x\rangle }\mu (\text{ d}x). \end{aligned}$$

For each Lévy process in law $X=\left\{ X_t,t\ge 0\right\}$ there exists a modification $Y=\left\{ Y_t,t\ge 0\right\}$ with right-continuous sample paths in $t\ge 0$, having left limits in $t>0$ and satisfying $P(X_t\ne Y_t)=0$, $t\ge 0$.

3.2 Self-Decomposable Lévy Processes

Definition 3.5

A probability distribution $\mu$ on $R^d$ is called self-decomposable, or of class $L(R^d)$, if, for any $b>1$, there exists a probability measure $\rho _b$ on $R^d$ such that

$$\begin{aligned} \hat{\mu }(z)=\hat{\mu }(b^{-1}z)\hat{\rho }_b(z), \quad z\in R^d, \end{aligned}$$

(3.2)

where $\hat{\mu }$ means the characteristic function of the probability distribution $\mu$ on $R^d$.

If $\mu$ is self-decomposable, then $\mu$ is infinitely divisible and, for any $b>1$, $\rho _b$ in the decomposition (3.2) is uniquely determined and $\rho _b$ is infinitely divisible.

Definition 3.6

A Lévy process $X=\left\{ X_t, t\ge 0\right\}$ in law is said to be self-decomposable if the probability distribution $\fancyscript{L}(X_1)$ is self-decomposable.

The Gaussian Lévy processes in law are, obviously, self-decomposable, because in this case (3.2) is satisfied with

$$\begin{aligned} \hat{\rho }_b(z)=\exp \left\{ i\langle z,(1-b^{-1})a\rangle -\frac{1}{2}\langle z(1-b^{-2})A,z\rangle \right\} . \end{aligned}$$

A criterion of self-decomposability of non-Gaussian $\mu \in ID(R^d)$ with the triplet $(a, A, \Pi )$ of Lévy characteristics will be formulated using the canonical polar decomposition of a Lévy measure$\Pi$ (see Remark 16 in [4, Lemma 1 in [5] and Proposition 2 in [6]).

Write

$$\begin{aligned} S^{d-1}=\left\{ x\in R^d:|x|=1\right\} , \quad K=\int \limits _{R_0^d}|x|^2\wedge 1\Pi (\text{ d}x)>0. \end{aligned}$$

Proposition 3.7

There exists a pair$(\lambda , \Pi _{\xi })$, where$\lambda$is a probability measure on$S^{d-1}$and$\Pi _{\xi }$is a$\sigma$-finite measure on$(0,\infty )$such that$\Pi _{\xi }(C)$is measurable in$\xi \in S^{d-1}$for every$C\in \fancyscript{B}\left((0,\infty )\right)$,

$$\begin{aligned} \int \limits ^{\infty }_{0}r^2\wedge 1\Pi _{\xi }(\text{ d}r)\equiv K \end{aligned}$$

(3.3)

and

$$\begin{aligned} \Pi (B)=\int \limits _{S^{d-1}}\lambda (\text{ d}\xi )\int \limits ^{\infty }_{0}1_B(r\xi )\Pi _{\xi }(\text{ d}r), \quad B\in {\fancyscript{B}}(B_{0}^{d}). \end{aligned}$$

(3.4)

If a pair$(\lambda ^{\prime },\Pi ^{\prime }_{\xi })$satisfies (3.3) and (3.4), then$\lambda ^{\prime }=\lambda$and$\Pi _{\xi }=\Pi ^{\prime }_{\xi }$$\lambda$-a.e.

Proof

Existence. Consider the probability space $(R_0^d, \fancyscript{B}(R_0^d),\mathrm P _{\Pi })$, where

$$\begin{aligned} \mathrm P _{\Pi }(B)=K^{-1}\int \limits _{B}|x|^2\wedge 1 \Pi (\text{ d}x), \quad B\in \fancyscript{B}(R_0^d). \end{aligned}$$

Let $N(x)=x$, $R(x)=|x|$, $\Xi (x)=\frac{x}{|x|}$, $x\in R_0^d$, $\lambda (B)=\mathrm P _{\Pi }\{\Xi \in B\}$, $B\in \fancyscript{B}(S^{d-1})$, $\Pi ^0_{\xi }(C)=\mathrm P _{\Pi }\{R\in C|\Xi =\xi \}$ (a regular version of the conditional distribution), and

$$\begin{aligned} \Pi _{\xi }(C)=\int \limits _{C}K(r^2\wedge 1)^{-1}\Pi ^0_{\xi }(\text{ d}r), \quad C\in \fancyscript{B}\left((0,\infty )\right), \quad \xi \in S^{d-1}. \end{aligned}$$

The pair $(\lambda ,\Pi _{\xi })$ satisfies (3.3) and (3.4). Indeed,

$$\begin{aligned} \int \limits ^{\infty }_{0}(r^2\wedge 1)\Pi _{\xi }(\text{ d}r)=K\int \limits ^{\infty }_{0}\Pi ^0_{\xi }(\text{ d}r)\equiv K, \end{aligned}$$

and for every nonnegative measurable function $f(x)$, $x\in R_0^d$,

$$\begin{aligned} \int \limits _{S^{d-1}}\lambda (\text{ d}\xi )\int \limits ^{\infty }_{0}f(r\xi )\Pi _{\xi }(\text{ d}r)&=\int \limits _{S^{d-1}}\lambda (\text{ d}\xi )\int \limits ^{\infty }_{0}\frac{Kf(r\xi )}{r^2\wedge 1}\Pi ^0_{\xi }(\text{ d}r)\\&=\mathrm E _{\Pi }\left[\mathrm E _{\Pi }\left(\frac{Kf(R\Xi )}{R^2 \wedge 1}|\Xi \right)\right] \\&= \mathrm E _{\Pi }\left(\frac{Kf(N)}{R^2\wedge 1}\right)=\int \limits _{R_0^d}f(x)\Pi (\text{ d}x). \end{aligned}$$

It remains to take $f(x)=1_B(x)$, $B\in \fancyscript{B}(R_0^d)$.

Uniqueness. Let

$$\begin{aligned} \Pi (B)=\int \limits _{S^{d-1}}\lambda ^{\prime }(\text{ d}\xi )\int \limits ^{\infty }_{0}1_B(r\xi )\Pi ^{\prime }_{\xi }(\text{ d}r) \end{aligned}$$

(3.5)

and

$$\begin{aligned} \int \limits ^{\infty }_{0}(r^2\wedge 1)\Pi ^{\prime }_{\xi }(\text{ d}r)\equiv K. \end{aligned}$$

(3.6)

Then, for all $B\in \fancyscript{B}(S^{d-1})$, from (3.3)–(3.6) we find that

$$\begin{aligned}&\int \limits _{R_0^d}1_B\left(\frac{x}{|x|}\right)K^{-1}(|x|^2\wedge 1)\Pi (\text{ d}x)\\&\qquad =\int \limits _{S^{d-1}}\lambda (\text{ d}\xi )\int \limits ^{\infty }_{0}1_B(\xi )K^{-1}(r^2\wedge 1)\Pi _{\xi }(\text{ d}r)=\lambda (B) \end{aligned}$$

and

$$\begin{aligned}&\int \limits _{R_0^d}1_B\left(\frac{x}{|x|}\right)K^{-1}(|x|^2\wedge 1)\Pi (\text{ d}x)\\&\qquad =\int \limits _{S^{d-1}}\lambda ^{\prime }(\text{ d}\xi )\int \limits ^{\infty }_{0}1_B(\xi )K^{-1}(r^2\wedge 1)\Pi ^{\prime }_{\xi }(\text{ d}r)=\lambda ^{\prime }(B), \end{aligned}$$

proving that $\lambda =\lambda ^{\prime }$.

Finally, for every nonnegative measurable function $h(r)$, $r>0$,

$$\begin{aligned} \int \limits _{R^d_0}h(|x|)\Pi (\text{ d}x)=\int \limits _{S^{d-1}}\lambda (\text{ d}\xi )\int \limits ^{\infty }_{0}h(r)\Pi _{\xi }(\text{ d}r)=\int \limits _{S^{d-1}}\lambda (\text{ d}\xi )\int \limits ^{\infty }_{0}h(r)\Pi ^{\prime }_{\xi }(\text{ d}r), \end{aligned}$$

implying that $\Pi _{\xi }=\Pi ^{\prime }_{\xi }$ $\lambda$-a.e. $\square$

Proposition 3.8

[7]. If

$$\begin{aligned} \Pi (B)=\int \limits _{B}g(x)\text{ d}x,\quad B\in \fancyscript{B}(R_0^d), \end{aligned}$$

(3.7)

then (3.3) and (3.4) hold with

$$\begin{aligned} \lambda (\text{ d}\xi )&=c(\xi )\text{ d}\xi ,\\ \Pi _{\xi }(\text{ d}r)&=r^{d-1}g(r\xi )c^{-1}(\xi ), \end{aligned}$$

where

$$\begin{aligned} c(\xi )=K^{-1}\int \limits ^{\infty }_{0}(r^2\wedge 1)r^{d-1}g(r\xi )\text{ d}r, \end{aligned}$$

assuming that

$$\begin{aligned} K:=\int \limits _{R_0^d}(|x|^2\wedge 1)g(x)\text{ d}x>0. \end{aligned}$$

Proof

Write

$$\begin{aligned}\begin{array}{rcl} x_1&=& r\cos \varphi _1,\\x_2 &=&r\sin \varphi _1\cos \varphi _2,\\&\cdots \cdots \cdots \cdots \\x_{d-1}&=& r\sin \varphi _1\sin \varphi _2\cdots \sin \varphi _{d-1}\cos \varphi _{d-1},\\ x_{d}&=& r\sin \varphi _1\sin \varphi {2}\cdots \sin \varphi _{d-2}\sin \varphi _{d-1},\end{array} \end{aligned}$$

where $r\ge 0$, $0\le \varphi _1\le \pi ,$ $\ldots$, $0\le \varphi _{d-2}\le \pi$, $0\le \varphi _{d-1}<2\pi$. It is well-known that the Jacobian

$$\begin{aligned} J=\frac{D(x_1,x_2,\ldots ,x_d)}{D(r,\varphi _1,\varphi _2,\ldots ,\varphi _{d-1})}=r^{d-1}\sin ^{d-2}\varphi _{1}\sin ^{d-3}\varphi _2\cdots \sin \varphi _{d-2}. \end{aligned}$$

Denoting $\xi =\frac{x}{r}$ and $\text{ d}\xi =\sin ^{d-2}\varphi _1\sin ^{d-3}\varphi _{2}\cdots \sin \varphi _{d-2}\text{ d}\varphi _1\text{ d}\varphi _2\cdots \text{ d}\varphi _{d-2}\text{ d}\varphi _{d-1}$, for any Borel measurable and integrable with respect to the Lebesgue measure on $R^d$ function $f(x)$, we find that

$$\begin{aligned} \int \limits _{R^d}f(x)\text{ d}x=\int \limits _{S^{d-1}}\text{ d}\xi \int \limits ^{\infty }_{0}f(r\xi )r^{d-1}\text{ d}r=\int \limits _{S^{d-1}}c(\xi )\text{ d}\xi \int \limits ^{\infty }_{0}f(r\xi )r^{d-1}c^{-1}(\xi )\text{ d}r \end{aligned}$$

(3.8)

and apply formula (3.8) to the functions $f_{B}(x)=g(x)1_B(x)$, $x\in R^d$, $B\in \fancyscript{B}(R_0^d)$. The identity (3.3) is trivially satisfied. $\square$

The following criterion of self-decomposability of a probability distribution $\mu \in ID (R^d)$ with the triplet of Lévy characteristics $(a, A, \Pi )$ is well-known (see Theorem 15.10 of [2] and [8]).

Theorem 3.9

A probability distribution$\mu \in ID (R^d)$or a Lévy process in law with the triplet$(a, A, \Pi )$of Lévy characteristics is self-decomposable if and only if in (3.4)

$$\begin{aligned} \Pi _{\xi }(\text{ d}r)=\frac{k_{\xi }(r)}{r}\text{ d}r, \end{aligned}$$

where a nonnegative function$k_{\xi }(r)$is measurable in$\xi \in S^{d-1}$and decreasing in$r>0$for$\lambda$-a.e. $\xi$.

Corollary 3.10

If (3.7) is satisfied, then$\mu \in ID(R^d)$or a corresponding Lévy process in law with the triplet$(a, A, \Pi )$of Lévy characteristics is self-decomposable if and only if the function$k_{\xi }(r):=r^dg(r\xi )$is decreasing in$r>0$for a.e.$\xi \in S^{d-1}$with respect to the Lebesgue surface measure on$S^{d-1}$.

3.3 Lévy Subordinators

Definition 3.11

An univariate Lévy process with nonnegative increments is called a Lévy subordinator.

The class of Lévy subordinators correspond one-to-one with the class $ID(R_+)$ of infinitely divisible distributions on $R_+$. It is well-known (see, e.g., [2, 3, 9, 10]) that for $\tau \in ID(R_+)$ the Laplace exponent

$$\begin{aligned} \psi (\theta ):=-\log \left(\int \limits ^{\infty }_{0}e^{-\theta u}\tau (\text{ d}u)\right)=\beta _0\theta +\int \limits ^{\infty }_{0}\left(1-e^{-\tau u}\rho (\text{ d}u)\right), \quad \theta \ge 0, \end{aligned}$$

is defined uniquely by the characteristics $(\beta _0,\rho )$, where $\beta _0\ge 0$ and $\rho$ is a $\sigma$-finite measure on $(0,\infty )$, satisfying

$$\begin{aligned} \int \limits ^{\infty }_{0}(u\wedge 1)\rho (\text{ d}u)<\infty . \end{aligned}$$

Extending the Thorin class and following Bondesson [11], we introduce the scale of Thorin classes $T_{\varkappa }(R_+)$, $0<\varkappa \le \infty$, as increasing subclasses of $ID(R_+)$ such that $T_{\infty }(R_+)=ID(R_+)$, where $T_{\infty }(R_+)$ is the minimal class of probability distributions on $R_+$, closed under convolutions and weak limits, containing all classes $T_{\varkappa }(R_+)$, $\varkappa >0$.

Definition 3.12

An infinitely divisible distribution $\tau$ on $R_+$ with the characteristics $(\beta _0,\rho )$ is of the Thorin class $T_{\varkappa }(R_+)$, $\varkappa >0$, if $\rho (\text{ d}t)=l(t)\text{ d}t$ and $k_{\varkappa }(t):=t^{2-\varkappa }l(t)$, $t\ge 0$, is completely monotone, i.e., $k_{\varkappa }$ is infinitely differentiable and $(-1)^nk^{(n)}_{\varkappa }(t)\ge 0$ for all $n\ge 0$ and $t>0$.

Lévy subordinators corresponding to the distributions from $T_{\varkappa }(R_+)$, $0<\varkappa <\infty$, are called the Thorin’s subordinators.

According to Bernstein’s theorem (see, e.g., [12]) there exists a unique positive measure $Q_{\varkappa }$ on $R_+$ such that

$$\begin{aligned} k_{\varkappa }(t)=\int \limits ^{\infty }_{0}e^{-vt}Q_{\varkappa }(\text{ d}v), \quad t>0, \end{aligned}$$

and

$$\begin{aligned} Q_{\varkappa }(\{0\})=\lim \limits _{t\rightarrow \infty }k_{\varkappa }(t). \end{aligned}$$

Write

$$\begin{aligned} a_{\varkappa }(t)=t^{-\varkappa }\int \limits ^{t}_{0}v^{\varkappa -1}e^{-v}\text{ d}v+t^{-\varkappa +1}\int \limits ^{\infty }_{t}v^{\varkappa -2}e^{-v}\text{ d}v, \quad t>0, \end{aligned}$$

and observe that, as $t\rightarrow \infty$,

$$\begin{aligned} a_{\varkappa }(t)\sim \Gamma (\varkappa )t^{-\varkappa }, \end{aligned}$$

and

$$\begin{aligned} a_{\varkappa }(t)\sim \left\{ \begin{array}{l@{\quad }l} (\varkappa (1-\varkappa ))^{-1},&\mathrm if \quad 0<\varkappa <1, \\ \log \dfrac{1}{t},&\mathrm if \quad \varkappa =1, \\ \Gamma (\varkappa -1)t^{1-\varkappa },&\mathrm if \quad \varkappa >1, \end{array} \right. \end{aligned}$$

(3.9)

as $t\rightarrow 0$.

Proposition 3.13

An infinitely divisible distribution$\tau$on$R_+$with the characteristics$(\beta _0, \rho )$is of the Thorin class$T_{\varkappa }(R_+)$, $\varkappa >0$, if and only if the Laplace exponent

$$\begin{aligned} \psi _{\varkappa }(\theta )=\left\{ \begin{array}{l} \beta _0\theta +\Gamma (\varkappa -1)\int \limits ^{\infty }_{0}\left(v^{-\varkappa +1}-(\theta +v)^{-\varkappa +1}\right)Q_{\varkappa }(\text{ d}v), \quad \mathrm if \quad \varkappa \ne 1, \\ \beta _0\theta +\int \limits ^{\infty }_{0}\log \left(1+\frac{\theta }{v}\right)Q_1(\text{ d}v), \quad \mathrm if \quad \varkappa =1, \end{array} \right. \end{aligned}$$

(3.10)

where the measure$Q_{\varkappa }$, called the Thorin measure, satisfies

$$\begin{aligned} \int \limits ^{\infty }_{0}a_{\varkappa }(v)Q_{\varkappa }(\text{ d}v)<\infty , \end{aligned}$$

(3.11)

implying that$Q_{\varkappa }(\{0\})=0$for$\varkappa \ge 1$.

Proof

We have that

$$\begin{aligned} \int \limits ^{\infty }_{0}(t\wedge 1)l(t)\text{ d}t=\int \limits ^{\infty }_{0}(t\wedge 1)t^{\varkappa -2}\int \limits ^{\infty }_{0}e^{-tv}Q_{\varkappa }(\text{ d}v)\text{ d}t=\int \limits ^{\infty }_{0}a_{\varkappa }(v)Q_{\varkappa }(\text{ d}v) \end{aligned}$$

and

$$\begin{aligned} \psi _{\varkappa }(\theta )&=\beta _0\theta +\int \limits ^{\infty }_{0}\left(1-e^{-\theta t}\right)t^{\varkappa -2}\int \limits ^{\infty }_{0}e^{-vt}Q_{\varkappa }(\text{ d}v)\text{ d}t\\&=\beta _0\theta +\int \limits ^{\infty }_{0}\left(\int \limits ^{\infty }_{0}t^{\varkappa -2}\left(1-e^{-\theta t}\right)e^{-vt}\text{ d}t\right)Q_{\varkappa }(\text{ d}v). \end{aligned}$$

However, for $0<\varkappa <1$,

$$\begin{aligned} \int \limits ^{\infty }_{0}t^{\varkappa -2}\left(1-e^{-\theta t}\right)e^{-vt}\text{ d}t&=-\int \limits ^{\infty }_{0}e^{-vt}t^{\varkappa -2}\sum \limits ^{\infty }_{k=1}\frac{(-t\theta )^k}{k!}\text{ d}t \\&= -\sum \limits ^{\infty }_{k=1}\frac{(-\theta )^k}{k!}v^{-k-\theta +1}\Gamma (k+\varkappa -1)\\&=-v^{-\varkappa +1}\sum \limits ^{\infty }_{k=1}\frac{\Gamma (k+\varkappa -1)}{k!}\left(-\frac{\theta }{v}\right)^k \\&=v^{-\varkappa +1}\Gamma (\varkappa -1)\left(1-\left(1+\frac{\theta }{v}\right)^{-\varkappa +1}\right), \end{aligned}$$

for $\varkappa =1$, as a Froullani integral,

$$\begin{aligned} \int \limits ^{\infty }_{0}\left(1-e^{-\theta t}\right)t^{-1}e^{-vt}\text{ d}t=\log \left(1+\frac{\theta }{v}\right) \end{aligned}$$

and, for $\varkappa >1$,

$$\begin{aligned} \int \limits ^{\infty }_{0}t^{\varkappa -2}\left(1-e^{-\theta t}\right) e^{-vt} \text{ d}t&=\frac{1}{\varkappa -1}\int \limits ^{\infty }_{0}\left(1-e^{-\theta t}\right)e^{-vt} \text{ d}t^{\varkappa -1} \\&=\frac{1}{\varkappa -1}\int \limits ^{\infty }_{0}t^{\varkappa -1}\left[ve^{-vt}-(\theta +v)e^{-(\theta +v)t}\right]\text{ d}t\\&=\frac{\Gamma (\varkappa )}{\varkappa -1}\left(v^{-\varkappa +1}-(\theta +v)^{-\varkappa +1}\right)\\&=\Gamma (\varkappa -1)\left(v^{-\varkappa +1}-(\theta +v)^{-\varkappa +1}\right). \end{aligned}$$

$\square$

Remark 3.14

Having in mind (3.9), inequality (3.11) is satisfied if and only if the measure $Q_{\varkappa }$ is a Radon measure such that for $\varkappa \ne 1$

$$\begin{aligned} \int \limits ^1_{0}u^{0\wedge (1-\varkappa )}Q_{\varkappa }(\text{ d}u)<\infty \quad \mathrm and \quad \int \limits ^{\infty }_{1}u^{-\varkappa }Q_{\varkappa }(\text{ d}u)<\infty , \end{aligned}$$

and for $\varkappa =1$

$$\begin{aligned} \int \limits ^{1}_{0}\log \left(\frac{1}{u}\right)Q_1(\text{ d}u)<\infty \quad \mathrm and \quad \int \limits ^{\infty }_{1}u^{-1}Q_1(\text{ d}u)<\infty . \end{aligned}$$

Recall now that the families of Tweedie or power-variance distributions

$$\begin{aligned} \left\{ Tw_p(\alpha ,\lambda ),\alpha >0,\lambda >0\right\} , \quad p\in R^1\backslash [0,1), \end{aligned}$$

are defined as exponential dispersion models (see [13–15]), satisfying the following properties: for each $\alpha >0$, $\lambda >0$ and given $p$

$$\begin{aligned} \int \limits _{R^1}xTw_{p}(\alpha ,\lambda )(\text{ d}x)&=\alpha ,\\ \int \limits _{R^1}(x-\alpha )^2Tw_p(\alpha ,\lambda )(\text{ d}x)&=\lambda ^{-1}\alpha ^p \end{aligned}$$

and $Tw_0(\alpha ,\lambda ):=N(\alpha ,\lambda ^{-1})$, $\alpha \in R^1$, $\lambda >0$. It is known that for $p\ge 1$ $Tw_p(\alpha ,\lambda )\in ID(R_+)$. Moreover, for $p>1$, $\alpha >0$, $\lambda >0$ $Tw_p(\alpha ,\lambda )\in T_{\frac{1}{p-1}}(R_+)$, because their characteristics are

$$\begin{aligned} \left(0,c_{p,\lambda }t^{-2+\frac{1}{p-1}}\exp \left\{ -\frac{\alpha ^{1-p}}{p-1}\lambda t\right\} \text{ d}t\right), \end{aligned}$$

where

$$\begin{aligned} c_{p,\lambda }=\frac{\lambda ^{\frac{1}{p-1}}}{\Gamma \left(\frac{p}{p-1}\right)(p-1)^{\frac{p}{p-1}}}, \end{aligned}$$

and $Tw_1(\alpha ,\lambda )\in ID(R_+)$ with characteristics $(0,\alpha \lambda \varepsilon _{\lambda ^{-1}}(\text{ d}t))$. The Thorin measure $Q_{\frac{1}{p-1}}$ of $Tw_p(\alpha ,\lambda )$, $p>1$, obviously, equals

$$\begin{aligned} c_{p,\lambda }\varepsilon _{\frac{\alpha ^{1-p}}{p-1}}(\text{ d}t). \end{aligned}$$

Theorem 3.15

[16].

• (i) Thorin classes$T_{\varkappa }(R_+)$, $0<\varkappa <\infty$, are increasing, closed under convolutions and weak limits;

• (ii) $T_{\infty }(R_+)=ID(R_+)$;

• (iii) Thorin classes$T_{\varkappa }(R_+)$, $0<\varkappa \le \infty$, are generalized convolutions of Tweedie distributions$T_{\frac{\varkappa +1}{\varkappa }}(\alpha ,\lambda )$, $\alpha >0$, $\lambda >0$.

Proof

Because for $0<\varkappa _1<\varkappa _2$

$$\begin{aligned} k_{\varkappa _2}(t)=t^{\varkappa _1-\varkappa _2}k_{\varkappa _2}(t), \quad t>0, \end{aligned}$$

$t^{-\gamma }$, $t>0$, $\gamma >0$, are completely monotone functions and the complete monotonicity is preserved under multiplication, from Definition 3.11 it follows that $T_{\varkappa _1}(R_+)\subset T_{\varkappa _2}(R_+)$.

Closedness of $T_{\varkappa }(R_+)$ under convolutions and weak limits follows from the well-known properties that the complete monotonicity is preserved under formation of linear combinations and pointwise limits (see, e.g., [12]).

• (ii) Observe that the characteristics and the Laplace exponent of $Tw_{\frac{\varkappa +1}{\varkappa }}(\alpha ,\lambda )$ are equal

  $$\begin{aligned} \left(0,\frac{\lambda ^{\varkappa }\varkappa ^{1+\varkappa }}{\Gamma (1+\varkappa )}t^{-2+\varkappa }e^{-\varkappa \lambda \alpha ^{-\frac{1}{\varkappa }}t}\text{ d}t\right) \end{aligned}$$

  and

  $$\begin{aligned} \psi _{\frac{1+\varkappa }{\varkappa },\alpha ,\lambda }(\theta )=\frac{\lambda \varkappa }{\varkappa ^{1-\varkappa }(\varkappa -1)}\left[\alpha ^{\frac{\varkappa -1}{\varkappa }}\varkappa ^{1-\varkappa }-\left(\varkappa \alpha ^{-\frac{1}{\varkappa }}+\frac{\theta }{\lambda }\right)^{1-\varkappa }\right] \end{aligned}$$

  (3.12)

  Because for each $\theta \ge 0$

  $$\begin{aligned} \lim \limits _{\varkappa \rightarrow \infty }\psi _{\frac{1+\varkappa }{\varkappa },\alpha ,\lambda }(\theta )&= \lim \limits _{\varkappa \rightarrow \infty }\frac{\lambda \varkappa \alpha ^{\frac{\varkappa -1}{\varkappa }}}{\varkappa -1}\left[1-\left(\frac{\theta }{\varkappa \lambda }\alpha ^{\frac{1}{\varkappa }}\right)\right]^{1-\varkappa }\nonumber \\&=\alpha \lambda \left(1-e^{-\frac{\theta }{\lambda }}\right), \end{aligned}$$

  (3.13)

  it follows that for all $\alpha >0$, $\lambda >0$ the scaled Poisson distributions $Tw_1(\alpha ,\lambda )\in T_{\infty }(R_+)$. Hawing in mind properties (i), we conclude from (3.13) that $T_{\infty }(R_+)=ID(R_+)$.

• (iii) The case $\varkappa =\infty$ is contained in (ii).

Let $0<\varkappa <\infty$. The statement (iii) follows easily from the Proposition 3.13, the formula (3.12) and the statement (i). $\square$

Remark 3.16

Because

$$\begin{aligned} Tw_2(\alpha ,\lambda )(\text{ d}t)=\frac{(\lambda \alpha ^{-1})^{\lambda }}{\Gamma (\lambda )}t^{\lambda -1}e^{-\lambda \alpha ^{-1}t}1_{(0,\infty )}\text{ d}t, \quad \text{ therefore} \quad \quad T_1(R_+)=GGC. \end{aligned}$$

Because

$$\begin{aligned} Tw_{\frac{3}{2}}(\alpha ,\lambda )(C)=e^{-2\lambda \sqrt{\alpha }}\varepsilon _0(C)+\int \limits _{C}\frac{2\lambda }{\sqrt{u}}e^{-2\frac{\lambda }{\sqrt{\alpha }}(u+\alpha )}I_1(4\alpha \sqrt{u})\text{ d}u, \quad C\in \fancyscript{B}(R_+), \end{aligned}$$

where $I_{\gamma }(z)$ is the modified Bessel function of the first kind (see Appendix), i.e.

$$\begin{aligned} I_{\gamma }(z)=\sum \limits ^{\infty }_{k=0}\frac{\left(\frac{z}{2}\right)^{2k+\gamma }}{k!\Gamma (\gamma +k+1)}, \quad \gamma \ge -1, \end{aligned}$$

is the compound Poisson-exponential distribution, therefore $T_2(R_+)$ is the class of generalized convolutions of compound Poisson-exponential distributions, which coincides with the generalized mixed exponential convolutions, studied by Goldie [17], Steutel [18, 19] and Bondesson [11].

Example 3.17

(noncentral gamma distribution) Following Fisher [20] (see also [21, 22]) we say that $\Gamma _{\beta ,\gamma ,\lambda }$ is a noncentral gamma distribution with the shape parameter $\beta >0$, the scale parameter $\gamma >0$ and the noncentrality parameter $\lambda >0$ if its pdf $f_{\beta ,\gamma ,\lambda }$ is the Poisson mixture of the gamma densities:

$$\begin{aligned} f_{\beta ,\gamma ,\lambda }(x)&=e^{-\lambda }\sum \limits ^{\infty }_{j=0}\frac{\lambda ^j}{j!}\frac{\beta ^{\gamma +j}x^{\gamma +j-1}}{\Gamma (\gamma +j)}e^{-\beta x}\\&=e^{-\lambda -\beta x}\beta \left(\frac{\beta x}{\lambda }\right)^{\frac{\gamma -1}{2}}I_{\gamma -1}\left(\sqrt{\beta \lambda x}\right), \quad x>0. \end{aligned}$$

Fisher in [20] derived that the probability law

$$\begin{aligned} \fancyscript{L}\left(\sum \limits ^{n}_{j=1}X^2_j\right)=\Gamma _{\frac{1}{2},\frac{n}{2},\lambda }, \end{aligned}$$

where $X_1,\ldots ,X_n$ are independent, $\fancyscript{L}(X_j)=N(\alpha _j,1)$ and $\lambda =\frac{1}{2}\sum \limits ^{n}_{j=1}\alpha ^2_j$.

Let $\text{ Bess}_{\beta ,\lambda }$, $\beta >0$, $\lambda >0$, be a probability distribution on $R_+$, defined by the formula:

$$\begin{aligned} \text{ Bess}_{\beta ,\lambda }(\text{ d}x)=e^{-\lambda }\varepsilon _0(\text{ d}x)+\beta e^{-\lambda -\beta x}I_1\left(2\sqrt{\beta \lambda x}\right)\text{ d}x. \end{aligned}$$

Because

$$\begin{aligned} \int \limits ^{\infty }_{0}e^{-\theta x}f_{\beta ,\gamma ,\lambda }(x)\text{ d}x&=e^{-\lambda }\beta (\beta \lambda )^{\gamma -1}\sum \limits ^{\infty }_{j=0}e^{-(\beta +\theta )x}\frac{(\beta \lambda x)^j}{j!\Gamma (\gamma +j)} \nonumber \\&=e^{-\lambda }\left(\frac{\beta }{\theta +\beta }\right)^{\gamma }\sum \limits ^{\infty }_{j=0}\frac{1}{j!}\left(\frac{\beta \lambda }{\theta +\beta }\right)^j\nonumber \\&=\left(\frac{\beta }{\theta +\beta }\right)^{\gamma }e^{\frac{-\lambda \theta }{\theta +\beta }} \end{aligned}$$

(3.14)

and

$$\begin{aligned} \int \limits ^{\infty }_{0}e^{-\theta x}\text{ Bess}_{\beta ,\lambda }(\text{ d}x)&=e^{-\lambda }+\lambda \beta e^{-\lambda }\sum \limits ^{\infty }_{k=0}\int \limits ^{\infty }_{0}e^{-(\theta +\beta )x}\frac{(\beta \lambda x)^k}{k!(k+1)!}\text{ d}x \nonumber \\&=e^{-\lambda }+e^{-\lambda }\sum \limits ^{\infty }_{k=0}\left(\frac{\beta \lambda }{\theta +\beta }\right)^{k+1}\frac{1}{(k+1)!}=e^{\frac{-\lambda \theta }{\theta +\beta }}, \end{aligned}$$

we find that

$$\begin{aligned} \Gamma _{\beta ,\gamma ,\lambda }=\Gamma _{\beta ,\gamma }*\text{ Bess}_{\beta ,\lambda }, \end{aligned}$$

implying equalities:

$$\begin{aligned} \Gamma _{\beta ,\gamma _1,\lambda _1}*\Gamma _{\beta ,\gamma _2,\lambda _2}&=\Gamma _{\beta ,\gamma _1+\gamma _2,\lambda _1+\lambda _2},\\ \Gamma _{\beta ,\gamma _1,\lambda }*\Gamma _{\beta ,\gamma _2}&=\Gamma _{\beta ,\gamma _1+\gamma _2,\lambda } \end{aligned}$$

and

$$\begin{aligned} \Gamma _{\beta ,\gamma ,\lambda _1}*\text{ Bess}_{\beta ,\lambda _2}=\Gamma _{\beta ,\gamma ,\lambda _1+\lambda _2}. \end{aligned}$$

From (3.14) it follows that

$$\begin{aligned} \int \limits ^{\infty }_{0}e^{-\theta x}f_{\beta ,\gamma ,\lambda }(x)\text{ d}x=\exp \left\{ \int \limits ^{\infty }_{0}\left(e^{-\theta u}-1\right)\left(\frac{\gamma }{u}+\lambda \right)e^{-\beta u} \text{ d}u\right\} , \end{aligned}$$

proving that the noncentral gamma distributions are infinitely divisible on $R_+$ with characteristics $(0,l_{\beta ,\gamma ,\lambda }(u)\text{ d}u)$, where

$$\begin{aligned} l_{\beta ,\gamma , \lambda }(u)=\left(\frac{\gamma }{u}+\lambda \right)e^{-\beta u}, \quad u>0. \end{aligned}$$

This function is completely monotone, implying that $\Gamma _{\beta , \gamma , \lambda }\in T_2(R_+)$.

Because the function

$$\begin{aligned} k_{\beta ,\gamma , \lambda }(u):=ul_{\beta ,\gamma ,\lambda }(u)=(\gamma +\lambda u)e^{-\beta u}, \quad u>0 \end{aligned}$$

(3.15)

is not completely monotone, $\Gamma _{\beta ,\gamma ,\lambda }\bar{\in }T_1(R_+)$. From (3.15) it follows that $k_{\beta ,\gamma ,\lambda }$ is nondecreasing if and only if $\lambda \le \beta \gamma$. Only in this case the noncentral gamma distribution $\Gamma _{\beta ,\gamma ,\lambda }$ is self-decomposable.

Inverse noncentral gamma distribution

$$\begin{aligned} I\Gamma _{\beta ,\gamma ,\lambda }(\text{ d}x):=x^{-2}f_{\beta ,\gamma ,\lambda }(x^{-1})\text{ d}x \end{aligned}$$

permits to define noncentral Student’s $t$-distribution $T_d(\nu ,\Sigma ,\alpha ,\lambda )$ with $\nu >0$ degrees of freedom, a scaling matrix $\Sigma$, a location vector $\alpha \in R^d$, and a noncentrality parameter $\lambda >0$ by means of the pdf $f_{\nu ,\Sigma ,\lambda }(x-\alpha )$, $x\in R^d$, where

$$\begin{aligned} f_{\nu ,\Sigma ,\lambda }(x)&=\int \limits ^{\infty }_{0}g_{0,u\Sigma }(x)u^{-2}f_{\frac{\nu }{2},\frac{\nu }{2},\lambda }\left(\frac{1}{u}\right)\text{ d}u\\&=\int \limits ^{\infty }_{0}g_{0,u\Sigma }(x)e^{-\lambda }\sum \limits ^{\infty }_{j=0}\frac{\lambda ^j}{j!}\frac{\left(\frac{\nu }{2}\right)^{\frac{\nu }{2}+j}}{\Gamma \left(\frac{\nu }{2}+j\right)}u^{-\frac{\nu }{2}-j-1}e^{-\frac{\nu }{2u}}\text{ d}u\\&= \frac{e^{-\lambda }\left(\frac{\nu }{2}\right)^{\frac{\nu }{2}}}{(2\pi )^{\frac{d}{2}}\sqrt{|\Sigma |}}\left(\frac{\nu }{2}+\frac{1}{2}\langle x\Sigma ^{-1}, x\rangle \right)^{-\frac{\nu +d}{2}}\\&\quad \times \sum \limits ^{\infty }_{j=0}\frac{\Gamma \left(\frac{\nu +d}{2}+j\right)}{j!\Gamma \left(\frac{\nu }{2}+j\right)}\left(\frac{\lambda \nu }{\nu +\langle x\Sigma ^{-1},x\rangle }\right)^j\\&= \frac{e^{-\lambda }}{(\nu \pi )\frac{d}{2}\sqrt{|\Sigma |}}\left(\frac{\nu +\langle x\Sigma ^{-1,x\rangle }}{\nu }\right)^{-\frac{\nu +d}{2}}\\&\quad \times \sum \limits ^{\infty }_{j=0}\frac{\Gamma \left(\frac{\nu +d}{2}+j\right)}{j!\Gamma \left(\frac{\nu }{2}+j\right)}\left(\frac{\lambda \nu }{\nu +\langle x\Sigma ^{-1},x\rangle }\right)^j. \end{aligned}$$

Analogously we define doubly noncentral Student’s $t$-distributions $T_d(\nu ,\Sigma ,\alpha ,a,\lambda )$ with $\nu >0$ degrees of freedom, a scaling matrix $\Sigma$, a location vector $\alpha \in R^d$, a noncentrality vector $a\in R^d$, and parameter $\lambda >0$ by means of pdf $f_{\nu ,\Sigma ,a,\lambda }(x-\alpha )$, $x\in R^d$, where

$$\begin{aligned} f_{\nu ,\Sigma ,a,\lambda }(x)&=\int \limits ^{\infty }_{0}g_{ua,u\Sigma }(x)u^{-2}f_{\frac{\nu }{2},\frac{\nu }{2},\lambda }\left(\frac{1}{u}\right)\text{ d}u\\&=\int \limits ^{\infty }_{0}g_{ua,u\Sigma }(x)e^{-\lambda }\sum \limits ^{\infty }_{j=0}\frac{\lambda ^j}{j!}\frac{\left(\frac{\nu }{2}\right)^{\frac{\nu }{2}+j}}{\Gamma \left(\frac{\nu }{2}+j\right)}u^{-\frac{\nu }{2}-j-1}e^{-\frac{\nu }{2u}}\text{ d}u\\&=\frac{e^{-\lambda }\left(\frac{\nu }{2}\right)^{\frac{\nu }{2}}2\exp \left\{ \langle x\Sigma ^{-1},x\rangle \right\} }{(2\pi )^{\frac{d}{2}}\sqrt{|\Sigma |}}\left(\frac{\langle a\Sigma ^{-1},a\rangle }{\nu +\langle x\Sigma ^{-1},x\rangle }\right)^{\frac{\nu +d}{4}}\\&\quad \times \sum \limits ^{\infty }_{j=0}\frac{\left(\frac{\lambda \nu }{2}\right)^j}{j!\Gamma \left(\frac{\nu }{2}+j\right)}\left(\frac{\langle a\Sigma ^{-1},a\rangle }{\nu +\langle x\Sigma ^{-1},x\rangle }\right)^{\frac{j}{2}}\\&\quad \times K_{\frac{\nu +d}{2}+j}\left(\left[\langle a\Sigma ^{-1},a\rangle \left(\nu +\langle a\Sigma ^{-1},x\rangle \right)\right]^{\frac{1}{2}}\right). \end{aligned}$$

Most likely, distributions $I\Gamma _{\beta ,\gamma ,\lambda }$, $T_d(\nu ,\Sigma ,\alpha ,\lambda )$ and $T_d(\nu ,\Sigma ,\alpha ,a,\lambda )$ are not infinitely divisible and do not correspond to any Lévy processes.

Example 3.18

(generalized gamma distribution). Recall that Bondesson introduced and studied in [11] a remarkable subclass of GGC of pdf on $(0,\infty )$, called the hyperbolically completely monotone pdf (HCM for short). It is said that $f$ is HCM, if for every $u>0$, $f(uv)f\left(\frac{u}{v}\right)$ is the completely monotone function in $w=v+v^{-1}$. For instance, GIG densities are HCM, because

$$\begin{aligned} gig(uv;\lambda ,\chi ,\psi )gig\left(\frac{u}{v};\lambda ,\chi ,\psi \right)=\frac{\left(\frac{\psi }{\chi }\right)^{\lambda }u^{2(\lambda -1)}}{\left(2K_{\lambda }(\sqrt{\chi \psi })\right)^2}\exp \left\{ -\frac{1}{2}\left(\chi u^{-1}+\psi u\right)w\right\} \end{aligned}$$

and the function $e^{-ax}$, $x>0$, $a>0$, is, obviously, completely monotone.

The generalized gamma density functions $g_{\beta ,\gamma ,\delta }$ are defined by the formula (see, e.g., [11]):

$$\begin{aligned} g_{\beta ,\gamma ,\delta }(x)=\frac{|\delta |}{\Gamma (\gamma )}\beta ^{\gamma }x^{\delta \gamma -1}\exp \left\{ -\beta x^{\delta }\right\} , \quad x>0, \quad \delta \in R^1_0, \quad \beta >0, \quad \gamma >0. \end{aligned}$$

It is proved in [11] that for $0<|\delta |\le 1$ $g_{\beta ,\gamma ,\delta }$ are HCM, because

$$\begin{aligned} g_{\beta ,\gamma ,\delta }(uv)g_{\beta ,\gamma ,\delta }\left(\frac{u}{v}\right)=\left(\frac{|\delta |\beta ^{\gamma }}{\Gamma (\gamma )}\right)^2u^{2(\delta \gamma -1)}\exp \left\{ -\beta u^{\delta }(v^{\delta }+v^{-\delta })\right\} \end{aligned}$$

and

$$\begin{aligned} \frac{\text{ d}}{\text{ d}w}\left(v^{\delta }+v^{-\delta }\right)=\frac{\delta \sin (\delta \pi )}{\pi }\int \limits ^{0}_{-\infty }\frac{|t|^{\delta }}{1+t^2-tw}\text{ d}t. \end{aligned}$$

The statement now follows from the known properties of completely monotone functions (see, e.g., [12]).

For $\delta >1$, pdf $g_{\beta ,\gamma ,\delta }$ are not infinitely divisible (see [11]) and, for $\delta <-1$, it is unknown whether or not $g_{\beta ,\gamma ,\delta }$ are infinitely divisible.

Following Definitions 1.1, 1.2 and using densities $g_{\beta ,\gamma ,\delta }$, $\delta <0$, $\beta >0$, $\gamma >0$, it is natural to define the generalized Student’s $t$-distributions with pdf as mixtures

$$\begin{aligned} \int \limits ^{\infty }_{0}g_{0,u\Sigma }(x-\alpha )g_{\beta ,\gamma ,\delta }(u)\text{ d}u, \quad x\in R^d \end{aligned}$$

and the generalized noncentral Student’s $t$-distributions with pdf as mixtures

$$\begin{aligned} \int \limits ^{\infty }_{0}g_{ua,u\Sigma }(x-\alpha )g_{\beta ,\gamma ,\delta }(u)\text{ d}u, \quad x\in R^d. \end{aligned}$$

In the case $-1\le \delta <0$ their pdf are infinitely divisible, but, excepting $\delta =-1$, their Lévy measure had no tractable expressions.

 

3.4 Subordinated Lévy Processes

Subordination of Markov processes as a transformation through random time change was introduced by Bochner in 1949 (see [23, 24]). In the context of Lévy processes subordination give us possibility to construct and investigate statistical models with desirable feature of the marginal distributions.

Let $X=\{X_t,t\ge 0\}$ be a Lévy process in $R^d$, $X_0\equiv 0$, with the triplet of Lévy characteristics$(a,A,\Pi )$ and the characteristic exponent

$$\begin{aligned} \varphi (z)&:=-\log Ee^{i\langle z,X_t\rangle }\\&=-i\langle a,z\rangle +\frac{1}{2}\langle zA,z\rangle -\int \limits _{R^d_0}\left(e^{i\langle z,x\rangle }-1-i\langle z,x\rangle 1_{\{|x|\le 1\}}\right)\Pi (\text{ d}x), \quad z\in R^d, \end{aligned}$$

called the subordinand process.

Let $T=\{T_t,t\ge 0\}$ be a Lévy subordinator, $T_0\equiv 0$, with the Laplace exponent

$$\begin{aligned} \psi (\theta ):=-\log \mathrm E e^{-\theta T_1}=\beta _0\theta +\int \limits ^{\infty }_{0}\left(1-e^{-\theta x}\right)\rho (\text{ d}x), \quad \theta \ge 0, \end{aligned}$$

and characteristics $(\beta _0,\rho )$, independent of $X$.

The subordinated process $\tilde{X}=\{\tilde{X}_t,t\ge 0\}$ is defined as a superposition

$$\begin{aligned} \tilde{X}_t=X_{T_t}, \quad t\ge 0. \end{aligned}$$

The following theorem is obtained by Zolotarev [25], Bochner [24], Ikeda and Watanabe [26], and Rogozin [27]. It was treated by Feller [28] and Sato [2]. These ideas were extended to the multivariate subordination of Lévy processes by Barndorff-Nielsen et al. in 2001 (see [5]).

Let

$$\begin{aligned} \mu ^t(B)&=\mathrm P \{X_t\in B\}, \quad B\in \fancyscript{B}(R^d), \quad t\ge 0,\\ \tau ^t(C)&=\mathrm P \{T_t\in C\}, \quad C\in \fancyscript{B}(R_+), \quad t\ge 0 \end{aligned}$$

and

$$\begin{aligned} \tilde{\mu }^t(B)=\mathrm P \{\tilde{X}_t\in B\}, \quad B\in \fancyscript{B}(R^d), \quad t\ge 0. \end{aligned}$$

 

Theorem 3.19

• (i) The subordinated process$\tilde{X}=\{\tilde{X}_t,t\ge 0\}$is a Lévy process with characteristic exponent$\tilde{\varphi }(z)=\psi \left(\varphi (z)\right)$, $z\in R^d$, and triplet of Lévy characteristics$(\tilde{a}, \tilde{A}, \tilde{\Pi })$, where

  $$\begin{aligned} \tilde{a}&=\beta _0a+\int \limits _{(0,\infty )}\left(\int \limits _{|x|\le 1}x\mu ^s(\text{ d}x)\right)\rho (\text{ d}s),\nonumber \\ \tilde{A}&=\beta _0A \end{aligned}$$

  (3.16)

  and

  $$\begin{aligned} \tilde{\Pi }(B)=\beta _0\Pi (B)+\int \limits _{(0,\infty )}\mu ^s(B)\rho (\text{ d}s), \quad B\in \fancyscript{B}(R_0^d). \end{aligned}$$

• (ii) For$t\ge 0$, $B\in \fancyscript{B}(R^d)$

  $$\begin{aligned} \tilde{\mu }^t(B)=\int \limits _{R_+}\mu ^s(B)\tau ^t(\text{ d}s). \end{aligned}$$

  (3.17)

We refer the reader for proof to [2].