On Natural Convergence Rate Estimates in the Lindeberg Theorem

Abstract

We prove two estimates of the rate of convergence in the Lindeberg theorem, involving algebraic truncated third-order moments and the classical Lindeberg fraction, which generalize a series of inequalities due to Esseen (Arkiv För Matematik 8, 1, 7–15, 1969), Rozovskii (Bulletin of Leningrad University (in Russian), (1):70–75, 1974), & Wang and Ahmad (Sankhya A: Indian J.Stat. 78, 2, 180–187, 2016), some of our recent results in Gabdullin, Makarenko, Shevtsova, (J. Math Sci. 234, 6, 847–885, 2018, J. Math Sci. 237, 5, 646–651, 2019b) and, up to constant factors, also Katz (Ann. Math. Statist. 34, 1107–1108, 1963), Petrov (Soviet Math. Dokl. 6, 5, 242–244, 1965), & Ibragimov and Osipov (Theory Probab. Appl. 11, 1, 141–143, 1966b). The technique used in the proof is completely different from that in Wang and Ahmad (Sankhya A: Indian J.Stat. 78, 2, 180–187, 2016) and is based on some extremal properties of introduced fractions which has not been noted in Katz (Ann. Math. Statist. 34, 1107–1108, 1963), Petrov (Soviet Math. Dokl. 6, 5, 242–244, 1965), & Wang and Ahmad (Sankhya A: Indian J.Stat. 78, 2, 180–187, 2016).

1 Introduction and Notation

Let X₁, X₂,…, X_n be independent random variables (r.v.’s) with distribution functions (d.f.’s) F_k(x) = P(X_k < x), $x\in \mathbb R,$ expectations EX_k = 0, and variances ${\sigma _{k}^{2}} = {\textsf {D}} X_{k}$, k = 1,…, n, such that

$$ {B_{n}^{2}}:= \sum\limits_{k=1}^{n} {\sigma_{k}^{2}}>0. $$

Let us denote

$$ S_{n} = X_{1} + X_{2} + {\cdots} + X_{n}, \quad \widetilde S_{n} = \frac{S_{n}-{\textsf{E}} S_{n}}{\sqrt{{\textsf{D}} S_{n}}} = \sum\limits_{k=1}^{n} \frac{X_{k}}{B_{n}}, $$

$$ \overline{F}_{n}(x) = {\textsf{P}}(\widetilde S_{n} < x), \quad {\Phi}(x) = \frac{1}{\sqrt{2\pi}} {\int}_{-\infty}^{x} e^{-t^{2}/2} dt, \quad x \in \mathbb R, $$

$$ {\Delta}_{n}= \sup\limits_{x \in \mathbb R} \left|\overline{F}_{n}(x) - {\Phi}(x)\right|, $$

$$ {\sigma_{k}^{2}}(z) = {\textsf{E}} {X_{k}^{2}} {\bf1}(\left|X_{k}\right| \ge z), \quad \mu_{k}(z) = {\textsf{E}} {X_{k}^{3}} {\bf1}(\left|X_{k}\right| < z),\quad k=1,\ldots,n $$

$$ L_{n}(z) = \frac{1}{{B_{n}^{2}}} \sum\limits_{k=1}^{n} {\sigma_{k}^{2}}(zB_{n}) = \frac{1}{{B_{n}^{2}}} \sum\limits_{k=1}^{n} {\textsf{E}} {X_{k}^{2}}{\bf1} (\left|X_{k}\right| \ge zB_{n}), \quad z \ge 0, $$

so that L_n(0) = 1. The function L_n(z) is called the Lindeberg fraction. In the case where X₁,…, X_n are independent and identically distributed (i.i.d.) we shall denote their common distribution function by F.

Let $\mathcal {G}$ be a set of all increasing functions $g\colon (0,\infty )\to (0,\infty )$ such that the function z/g(z) is also increasing for z > 0 (for convenience, here and in what follows, we use the terms “increasing” and “decreasing” in a wide sense, i. e. “non-decreasing”, “non-increasing”). Where appears, the value g(0) is assumed to be an arbitrary non-negative number. The class $\mathcal {G}$ was originally introduced by Katz (1963) and further used in Petrov (1965), Korolev and Popov (2012, 2017), Gabdullin et al. (2019b). In Gabdullin et al. (2019b) it was proved that:

• (i) For every $g \in \mathcal {G}$ and a > 0 we have

  $$ g_{0}(z, a):=\min\left\{\frac{z}{a}, 1\right\}\ \le\ \frac{g(z)}{g(a)}\ \le\ \max\left\{\frac{z}{a}, 1\right\}:= g_{1}(z, a),\quad z>0, $$

  (1.1)

  moreover g₀(⋅, a), $g_{1}(\cdot , a) \in \mathcal {G}.$

• (ii) Every function from $\mathcal {G}$ is continuous on $(0,\infty ).$

Property (i) means that, asymptotically, every function $g\in \mathcal {G}$ is between a constant and a linear function as its argument goes to infinity. For example, besides g₀, g₁, the class $\mathcal {G}$ also includes the following functions:

$$ g_{\textsc{c}}(z)\equiv 1,\quad g_{*}(z)=z,\quad c \cdot z^{\delta}, \quad c\cdot g(z),\qquad z>0, $$

for every c > 0, δ ∈ [0,1], and $g\in \mathcal {G}$.

For every function $g\in \mathcal {G}$ such that ${\textsf {E}} {X_{k}^{2}}g(\left |X_{k}\right |) < \infty ,$ k = 1,…, n, Katz (1963) (in 1963, for identically distributed random summands) and Petrov (1965) (in 1965, in the general situation) proved that

$$ {\Delta}_{n} \le \frac{A_{1}}{{B_{n}^{2}}g(B_{n})}\sum\limits_{k=1}^{n} {\textsf{E}} {X_{k}^{2}}g(\left|X_{k}\right|), $$

(1.2)

A₁ being a universal constant whose best known upper bound A₁ ≤ 1.87 is due to Korolev and Dorofeyeva (2017). A few years earlier, letting n = 1 and using results of Agnew (1957) (see also Bhattacharya and Ranga Rao 1976) and Kondrik et al. (2006) (see also Chebotarev et al. 2007), Korolev and Popov (2012) established a lower bound

$$ A_{1}\ge \sup\limits_{x>0}\left|\frac1{1+x^{2}}-{\Phi}(-x)\right|=0.54093\ldots, $$

which remains the best known one until now. For the sake of unambiguity, here and in what follows, by constants appearing in majorizing expressions we mean their least possible values guaranteeing the validity of the corresponding inequalities for all parameters under consideration.

Inequality (1.2) with $g(x) = {\min \limits } \{x/B_{n}, 1\}$ yields

$$ {\Delta}_{n} \le A_{2} \cdot (L_{n}(1) + {\Lambda}_{n}(1)) $$

(1.3)

where A₂ ≤ A₁ and

$$ {\Lambda}_{n}(\varepsilon) := \frac{1}{{B_{n}^{3}}}\sum\limits_{k=1}^{n} {\textsf{E}}|X_{k}|^{3}{\bf1}(|X_{k}|<\varepsilon B_{n}) = \frac{1}{{B_{n}^{3}}}\sum\limits_{k=1}^{n} {\int}_{\left|x\right| < \varepsilon B_{n}}\left|x\right|^{3} dF_{k}(x),\quad \varepsilon>0. $$

As it was pointed out in Loh (1975) (see also Paditz 1984),

$$ {\textsf{E}}|X|^{3}{\bf1}(|X|<1)+ {\textsf{E}} X^{2}{\bf1}(|X|\ge1)= {\textsf{E}} X^{2}\min\{ 1,|X|\}({\bf1}(X\in B) +{\bf1}(X\notin B)) $$

$$ \le {\textsf{E}}|X|^{3}{\bf1}(X \in B) + {\textsf{E}} X^{2}{\bf1}(X\notin B) $$

for any Borel $B\subseteq \mathbb R$, so that

$$ L_{n}(1) + {\Lambda}_{n}(1) = \inf_{\varepsilon > 0} (L_{n}(\varepsilon) + {\Lambda}_{n}(\varepsilon)), $$

and, hence, Eq. 1.3 yields the inequality

$$ {\Delta}_{n} \le A_{2} \cdot (L_{n}(\varepsilon) + {\Lambda}_{n}(\varepsilon)) \quad\text{for every }\varepsilon>0. $$

(1.4)

Inequality (1.4) was proved by Osipov (1966), independently of Eq. 1.2. Since

$$ {\Lambda}_{n}(\varepsilon)\le\frac{\varepsilon}{{B_{n}^{2}}}\sum\limits_{k=1}^{n}{\textsf{E}} {X_{k}^{2}}{\bf1}(|X_{k}|<\varepsilon B_{n})\le\varepsilon, $$

(1.5)

and, hence,

$$ {\Delta}_{n} \le A_{2} (L_{n}(\varepsilon) + \varepsilon) \quad\text{for every }\varepsilon>0, $$

inequality (1.4) trivially yields the Lindeberg theorem (Lindeberg, 1922) which states the sufficiency of the Lindeberg condition

for the validity of the CLT

On the other hand, for the asymptotically negligible random summands, i.e. satisfying the Feller condition

condition (CLT), according to Feller’s theorem (Feller, 1935), yields (L). Hence, the right- and the left-hand sides of Eq. 1.4 either tend or do not tend to zero simultaneously, once the random summands are uniformly asymptotically negligible in the sense of (F). Thus, using Zolotarev’s classification (Zolotarev, 1997), inequality (1.4) can be called a natural convergence rate estimate in the Lindeberg–Feller theorem.

In 1984 Paditz (1984) observed without proof that A₁ = A₂; in 2012 Korolev and Popov (2012) provided a complete proof of the equality A₁ = A₂. In other words, the function $g(x)=g_{0}(x,B_{n})=\min \limits \{x/B_{n}, 1\}$ minimizes the right-hand side of Eq. 1.2 (observe that this fact also trivially follows from property (i) of the functions $g\in \mathcal G$ (see Eq. 1.1) proved in Gabdullin et al. 2019b). So, inequalities (1.2), (1.3), and (1.4) are equivalent and, hence, can be called natural convergence rate estimates in the Lindeberg–Feller theorem.

On the other hand, inequality (1.2) with g(x) = x reduces to the celebrated Berry–Esseen inequality (Berry, 1941; Esseen, 1942) up to the constant factor A₁, for which Shevtsova (2013) provides an improved upper bound: A₁ ≤ 0.4690 in the i.i.d. case and A₁ ≤ 0.5583 in the general situation.

Esseen (1969) managed to replace the absolute truncated third order moments Λ_n(ε) in Osipov’s inequality (1.4) with absolute values of the algebraic ones and to prove that

$$ {\Delta}_{n} \le \frac{A_{3}}{{B_{n}^{3}}} \sum\limits_{k=1}^{n} \sup\limits_{z > 0} \left\{\left|\mu_{k}(z)\right|+z{\sigma_{k}^{2}}(z)\right\}, $$

(1.6)

with A₃ being an absolute constant. Moreover, in the same paper, using the traditional truncation techniques, Esseen provided a sketch of the proof of a bounded version of his inequality (1.6):

$$ {\Delta}_{n} \le \frac{A_{4}}{{B_{n}^{3}}} \sum\limits_{k=1}^{n} \sup\limits_{0 < z < B_{n}} \left\{\left|\mu_{k}(z)\right|+z{\sigma_{k}^{2}}(z)\right\}, $$

(1.7)

which trivially yields (1.3) with A₂ ≤ A₄, since |μ_k(z)|≤E|X_k|³1(|X_k| < z) and the function

$$ {\textsf{E}}|X_{k}|^{3}{\bf1}(|X_{k}|<z)+z{\sigma_{k}^{2}}(z)={\textsf{E}} {X_{k}^{2}}\min\{|X_{k}|,z\} $$

is monotonically increasing with respect to z ≥ 0 for every k = 1,…, n. Hence, inequality (1.7) is also a natural convergence rate estimate in the Lindeberg–Feller theorem and

$$ A_{2}\le A_{4},\quad A_{3}\le A_{4}. $$

Observe that, due to the left-continuity of the functions μ_k(z) and $\sigma ^{2}_{k}(z)$, k = 1,…, n, for z > 0, the least upper bound over z ∈ (0, B_n) in Eq. 1.7 can be replaced by the one over the set z ∈ (0, B_n]. In the i.i.d. case, Esseen’s inequality (1.7) takes the form

$$ {\Delta}_{n}\le \frac{A_{4}}{{\sigma_{1}^{3}}\sqrt{n}}\sup\limits_{0<z\le \sigma_{1}\sqrt{n}} \left\{\left|\mu_{1}(z)\right| + z\sigma^{2}_{1}(z)\right\} $$

and trivially yields the “if” part of Ibragimov’s criteria (Ibragimov, 1966a), according to which, ${\Delta }_{n}=\mathcal O(n^{-1/2})$ as $n\to \infty$ if and only if

$$ \mu_{1}(z)=\mathcal O(1),\quad z\sigma^{2}_{1}(z)=\mathcal O(1),\quad z\to\infty. $$

The values of A₃ and A₄ remained unknown for a long time. Only in 2018 the present authors (Gabdullin et al. 2018) proved that A₃ ≤ 2.66 and A₄ ≤ 2.73.

In 1974 Rozovskii (1974) proved that

$$ {\Delta}_{n} \le \frac{A_{5}}{{B_{n}^{3}}} \left( \left|\sum\limits_{k=1}^{n} \mu_{k}(B_{n})\right| + \sup\limits_{0 < z < B_{n}}\sum\limits_{k=1}^{n} z {\sigma_{k}^{2}}(z) \right), $$

(1.8)

where A₅ is an absolute constant whose value also remained unknown for a long time until the present authors deduced in Gabdullin et al. (2018) that A₅ ≤ 2.73. In Gabdullin et al. (2018, Section 5), it is also shown that Esseen’s and Rozovskii’s fractions (the right-hand sides of Eqs. 1.7 and 1.8, ignoring the constant factors A₄ and A₅) are incomparable even in the i.i.d. case, that is, Rozovskii’s fraction may be less and may be greater than Esseen’s fraction. Generally speaking, Rozovskii’s inequality (1.8) doesn’t yield Lindeberg’s theorem even in the symmetric case, while it’s improvement in Eq. 1.11 below does (in the symmetric case).

Adopting ideas of Katz (1963) and Petrov (1965), recently, Wang and Ahmad (2016) generalized Esseen’s inequality (1.6) to

$$ {\Delta}_{n} \le \frac{A_{6}}{{B_{n}^{2}}g(B_{n})} \sum\limits_{k=1}^{n} \sup\limits_{z > 0} \frac{g(z)}{z} \Big(\left|\mu_{k}(z)\right| + z{\sigma_{k}^{2}}(z) \Big), \quad g\in\mathcal{G}, $$

(1.9)

where A₆ is an absolute constant whose value has not been given in Wang and Ahmad (2016). One can make sure that inequality (1.9) trivially yields (1.2) with A₁ ≤ A₆ (for the complete proof, see Gabdullin et al. 2019b, p. 648) and with g(z) = z reduces to Eq. 1.6 with A₃ ≤ A₆. Thus, inequality (1.9) is also a natural convergence rate estimate in the Lindeberg–Feller theorem. In Gabdullin et al. (2019b) it was shown that A₆ ≤ 2.73.

Inequalities (1.6), (1.7), and (1.8) were improved and generalized in Gabdullin et al. (2018) to

$$ {\Delta}_{n} \le \frac{A_{\textsc{e}}(\varepsilon,\gamma)}{{B_{n}^{3}}} \sup\limits_{0 < z < \varepsilon B_{n}} \left\{ \gamma\left|\sum\limits_{k=1}^{n} \mu_{k}(z)\right| + z \sum\limits_{k = 1}^{n} {\sigma_{k}^{2}}(z) \right\} =: A_{\textsc{e}}(\varepsilon,\gamma) \cdot L_{\textsc{e},n}(\varepsilon,\gamma), $$

(1.10)

$$ {\Delta}_{n} \le \frac{A_{\textsc{r}}(\varepsilon,\gamma)}{{B_{n}^{3}}} \left( \gamma\left|\sum\limits_{k=1}^{n} \mu_{k}(\varepsilon B_{n})\right| + \sup\limits_{0 < z < \varepsilon B_{n}} z \sum\limits_{k=1}^{n} {\sigma_{k}^{2}}(z) \right) =: A_{\textsc{r}}(\varepsilon,\gamma) \cdot L_{\textsc{r},n}(\varepsilon,\gamma), $$

(1.11)

for every ε > 0 and γ > 0, where A_e(ε, γ), A_r(ε, γ) depend only on ε and γ, both are monotonically decreasing with respect to γ > 0, and A_e(ε, γ) is also monotonically decreasing with respect to ε > 0. In particular,

$$ A_{4}\le A_{\textsc{e}}(1,1) \le A_{\textsc{e}}(1,0.72) \le 2.73, $$

$$ A_{3}\le A_{\textsc{e}}(+\infty,1)\le\max\{A_{\textsc{e}}(+\infty,0.97),A_{\textsc{e}}(4.35,1)\} \le 2.66, $$

$$ A_{5}= A_{\textsc{r}}(1,1)\le A_{\textsc{r}}(1,\gamma_{*}) \le 2.73, $$

where

$$ \gamma_{*}=1/\sqrt{6\varkappa}=0.5599\ldots,\quad \varkappa=x^{-2}\sqrt{(\cos x-1+x^{2}/2)^{2}+(\sin x-x)^{2}}\Big|_{x=x_{0}}=0.5315\ldots, $$

and x₀ = 5.487414… is the unique root of the equation

$$ 8(\cos x - 1) + 8x\sin x - 4x^{2}\cos x - x^{3}\sin x = 0,\quad x\in(\pi,2\pi). $$

Moreover, the functions A_e(ε, γ) and A_r(ε, γ) are unbounded as ε → 0 + for every γ, since $\lim \limits _{\varepsilon \to 0}L_{{\textsc {e}},n}(\varepsilon ,\gamma )=\lim \limits _{\varepsilon \to 0}L_{{\textsc {r}},n}(\varepsilon ,\gamma )=0$. The question on boundedness of $A_{\textsc {r}}(+\infty ,\gamma )$ remains open. The values of A_e(ε, γ) and A_r(ε, γ) for some other ε and γ computed in Gabdullin et al. (2018) are presented in Tables 1 and 2, respectively. Observe that L_{R, n}(1,1) coincides with the Rozovskii fraction in Eq. 1.8 and, in the i.i.d. case, the fractions L_{e, n}(1,1) and $L_{{\textsc {e}},n}(\infty ,1)$ coincide with the Esseen fractions in Eqs. 1.6 and 1.7, respectively, so that

$$ A_{3}=A_{\textsc{e}}(\infty,1),\quad A_{4}=A_{\textsc{e}}(1,1),\quad\text{in the i.i.d. case.} $$

Table 1 Upper bounds for A_e(ε, γ) in Eq. 1.10

Table 2 Upper bounds for A_r(ε, γ) in Eq. 1.11

Using the notation

$$ M_{n}(z):=\frac1{{B_{n}^{3}}}\sum\limits_{k=1}^{n} \mu_{k}(zB_{n}) = \frac1{{B_{n}^{3}}}\sum\limits_{k=1}^{n} {\textsf{E}} {X_{k}^{3}}{\bf1}(|X_{k}|<zB_{n}),\quad z\ge0, $$

inequalities (1.10) and (1.11) can be rewritten as

$$ {\Delta}_{n} \le \inf_{\varepsilon,\gamma>0}A_{\textsc{e}}(\varepsilon,\gamma)\sup\limits_{0 < z < \varepsilon} \left\{ \gamma\left|M_{n}(z)\right| + zL_{n}(z) \right\} , $$

(1.12)

$$ {\Delta}_{n} \le \inf_{\varepsilon,\gamma>0}A_{\textsc{r}}(\varepsilon,\gamma)\Big(\gamma\left|M_{n}(\varepsilon)\right| + \sup\limits_{0 < z < \varepsilon} zL_{n}(z) \Big). $$

(1.13)

Let us show that inequality (1.12) already with ε = γ = 1 is a natural convergence rate estimate in the Lindeberg–Feller theorem. Indeed, the fraction in the right-hand side of Eq. 1.12 with ε = γ = 1 satisfies

$$ \sup\limits_{0<z<1} \left\{\left|M_{n}(z)\right| + zL_{n}(z) \right\}\le \sup\limits_{0<z<1} \left\{{\Lambda}_{n}(z) + zL_{n}(z) \right\} =\sup\limits_{0<z<1}\sum\limits_{k=1}^{n}{\textsf{E}} {X_{k}^{2}}\min\{|X_{k}|,z\} $$

$$ =\sum\limits_{k=1}^{n}{\textsf{E}} {X_{k}^{2}}\min\{|X_{k}|,1\}={\Lambda}_{n}(1) + L_{n}(1) = \inf_{\delta>0}\big\{{\Lambda}_{n}(\delta)+L_{n}(\delta) \big\}\le \inf_{\delta>0}\big\{\delta+L_{n}(\delta) \big\} $$

hence, under the Lindeberg condition, inequality (1.12) yields

$$ {\Delta}_{n}\le A_{\textsc{e}}(1,1)\inf_{\delta>0}\big\{\delta+L_{n}(\delta) \big\}\to A_{\textsc{e}}(1,1)\inf_{\delta>0}\delta=0, \quad n\to\infty, $$

i.e. Lindeberg’s theorem follows from Eq. 1.12.

Furthermore, inequality (1.13) is a natural convergence rate estimate in the Lindeberg–Feller theorem in case of existence of such an ε₀ > 0 that M_n(ε₀) = 0 for sufficiently large $n\in \mathbb N$ (in particular, in case of symmetric distributions of random summands, where one can take arbitrary ε₀ > 0). Indeed, the fraction in the right-hand side of Eq. 1.13 with ε = ε₀ for every δ ∈ (0, ε₀) satisfies

$$ \begin{array}{@{}rcl@{}} \gamma\left|M_{n}(\varepsilon_{0})\right| + \sup\limits_{0<z<\varepsilon_{0}} zL_{n}(z) &=& \max\Big\{\sup\limits_{0<z\le\delta}zL_{n}(z),\sup\limits_{\delta<z<\varepsilon_{0}}zL_{n}(z)\Big\}\\ &&\le \max\Big\{\delta L_{n}(0),\varepsilon_{0} L_{n}(\delta)\Big\}, \end{array} $$

and, under the Lindeberg condition, inequality (1.13) with ε = ε₀ and, say, γ = γ_∗ = 0.5599… yields

$$ {\Delta}_{n} \le A_{\textsc{r}}(\varepsilon_{0},\gamma_{*})\inf_{\delta>0}\max\Big\{\delta L_{n}(0),\varepsilon_{0} L_{n}(\delta)\Big\} \to A_{\textsc{r}}(\varepsilon_{0},\gamma_{*})\inf_{\delta>0}\delta =0, \quad n\to\infty, $$

i.e. Lindeberg’s theorem follows from Eq. 1.13 under the above condition M_n(ε₀) = 0, where ε₀ is independent of n.

Inequalities (1.9) and (1.10) with γ = 1 were generalized in our previous paper (Gabdullin et al. 2019b) to

$$ \begin{array}{@{}rcl@{}} {\Delta}_{n}\!\! &\!\le\!&\!\! \frac{C_{\textsc{e}}(\varepsilon)}{{B_{n}^{2}} g(B_{n})} \cdot \sup\limits_{0 < z < \varepsilon B_{n}} \frac{g(z)}{z} \left( \left|\sum\limits_{k=1}^{n} \mu_{k}(z)\right| + z \sum\limits_{k = 1}^{n} {\sigma_{k}^{2}}(z) \right)\\ \!\!& = &\!\! C_{\textsc{e}}(\varepsilon)\sup\limits_{0 < z < \varepsilon} \frac{g(zB_{n})}{zg(B_{n})} \left( \left|M_{n}(z)\right| + zL_{n}(z) \right), \quad \varepsilon>0, \quad g \in \mathcal{G}, \end{array} $$

(1.14)

with $C_{\textsc {e}}(\varepsilon )\le A_{\textsc {e}}(\min \limits \{1,\varepsilon \},1)$, $C_{\textsc {e}}(+0)=\infty ,$ so that $A_{6}\le C_{\textsc {e}}(\infty )$ with the equality in the i.i.d. case, and inequality (1.11) with γ = 1 was generalized in Gabdullin et al. (2019a) to

$$ \begin{array}{@{}rcl@{}} {\Delta}_{n} &\le& \frac{C_{\textsc{r}} (\varepsilon)}{{B_{n}^{2}} g(B_{n})} \left( \frac{g(\varepsilon B_{n})}{\varepsilon B_{n}} \left|\sum\limits_{k=1}^{n} \mu_{k}(\varepsilon B_{n})\right| + \sup\limits_{0 < z < \varepsilon B_{n}} g(z)\sum\limits_{k = 1}^{n} {\sigma_{k}^{2}}(z) \right)\\ &=& C_{\textsc{r}}(\varepsilon)\left( \frac{g(\varepsilon B_{n})}{\varepsilon g(B_{n})} \left|M_{n}(\varepsilon)\right| + \sup\limits_{0 < z < \varepsilon}\frac{g(zB_{n})}{g(B_{n})} L_{n}(z) \right), \quad \varepsilon>0, \quad g \in \mathcal{G},\\ \end{array} $$

(1.15)

with $C_{\textsc {r}}(\varepsilon )\le \max \limits \{1,\varepsilon \}\cdot A_{\textsc {r}}(\varepsilon ,1)$, $C_{\textsc {r}}(+0)=C_{\textsc {r}}(\infty )=\infty$. It is easy to see that Eqs. 1.14 and 1.15 with g(z) = g_∗(z)(≡ z) reduce, respectively, to Eqs. 1.10 and 1.11 with γ = 1 and A_e(⋅,1) ≤ C_e(⋅), A_r(⋅,1) ≤ C_r(⋅), so that

$$ C_{\textsc{e}}(\varepsilon)=A_{\textsc{e}}(\varepsilon,1),\quad C_{\textsc{r}}(\varepsilon)=A_{\textsc{r}}(\varepsilon,1)\quad\text{for } \varepsilon\in(0,1]. $$

Though the constants A₁ = A₂ ≤ 1.87 in Katz–Petrov’s (1.2) and Osipov’s (1.4) inequalities are more optimistic than A_e(1,1) = C_e(1) ≤ 2.73, C_r(1) = A_r(1,1) ≤ 2.73, inequalities (1.10), (1.11), (1.14), and (1.15) may be much sharper than Eqs. 1.2 and 1.4 due to the more favorable dependence of the appearing fractions on truncated third order moments $\left |M_{n}(\cdot )\right |$, which vanish, say, in the symmetric case, or for even n and oscillating sequence X_k=d(− 1)^kX, k = 1,…, n, with one and the same r.v. X. In the above cases inequalities (1.10) and (1.11) reduce to

$$ {\Delta}_{n} \le C_{\varepsilon}\cdot\sup\limits_{0 < z <\varepsilon} zL_{n}(z),\quad C_{\varepsilon}=\min\{A_{\textsc{e}}(\varepsilon,\infty),A_{\textsc{r}}(\varepsilon,\infty)\}. $$

Since L_n(z) is non-increasing and left-continuous, the least upper bound here must be attained in an interior of the interval (0, ε):

$$ {\Delta}_{n} \le C_{\varepsilon}\cdot z_{n}L_{n}(z_{n})\quad\text{with some } z_{n}\in(0,\varepsilon). $$

The sequence $\{z_{n}\}_{n\in \mathbb N}$ here may be infinitesimal as $n\to \infty$. This follows from Ibragimov and Osipov’s result (Ibragimov and Osipov, 1966b) who proved that, in general, the estimate Δ_n ≤ CL_n(z) cannot hold with a fixed z > 0, even in the symmetric i.i.d. case.

2 Motivation, Main Results and Discussion

As it was noted before, the sum of truncated third order moments M_n(⋅) in Eqs. 1.14 and 1.15 may be arbitrarily small or even vanish, so that the term depending on the Lindeberg fraction L_n may be much greater than the term containing M_n. Hence, it would be useful to have a possibility to balance the contribution of the terms $\left |M_{n}\right |$ and L_n to optimize the resulting bound. Similarly to Eqs. 1.10 and 1.11, let us introduce a balancing parameter γ > 0 and for ε > 0 and $g\in \mathcal {G}$ denote

$$ \begin{array}{@{}rcl@{}} L_{\textsc{e},n}(g,\varepsilon, \gamma) &=& \frac{1}{{B_{n}^{2}} g(B_{n})} \sup\limits_{0 < z < \varepsilon B_{n}} \frac{g(z)}{z} \Bigg\{ \gamma \left|\sum\limits_{k = 1}^{n} \mu_{k}(z)\right| + z\sum\limits_{k = 1}^{n}{\sigma_{k}^{2}}(z) \Bigg\}\\ &=& \sup\limits_{0 < z < \varepsilon} \frac{g(zB_{n})}{zg(B_{n})} \left( \gamma\left|M_{n}(z)\right| + zL_{n}(z) \right), \end{array} $$

(2.1)

$$ \begin{array}{@{}rcl@{}} L_{\textsc{r},n}(g,\varepsilon, \gamma) \!& = &\! \frac{1}{{B_{n}^{2}} g(B_{n})}\left( \gamma \frac{g(\varepsilon B_{n})}{\varepsilon B_{n}}\left|\sum\limits_{k=1}^{n} \mu_{k}(\varepsilon B_{n})\right|+\sup\limits_{0<z < \varepsilon B_{n}} g(z)\sum\limits_{k=1}^{n} {\sigma_{k}^{2}}(z)\right)\\ \!& = &\! \gamma \frac{g(\varepsilon B_{n})}{\varepsilon g(B_{n})} \left|M_{n}(\varepsilon)\right| + \sup\limits_{0 < z < \varepsilon}\frac{g(zB_{n})}{g(B_{n})} L_{n}(z). \end{array} $$

(2.2)

Then L_{e, n}(g, ε,1), L_{r, n}(g, ε,1) coincide with the corresponding fractions in the right-hand sides of Eqs. 1.14 and 1.15; moreover, with

$$ g(z)=g_{*}(z)\equiv z,\quad z>0, $$

we have

$$ L_{\textsc{e},n}(g_{*},\varepsilon,\gamma) = L_{\textsc{e},n}(\varepsilon,\gamma),\quad L_{\textsc{r},n}(g_{*},\varepsilon,\gamma) = L_{\textsc{r},n}(\varepsilon,\gamma), \quad \varepsilon,\gamma>0. $$

Observe also that L_{e, n}(⋅, ε,⋅) is monotonically increasing with respect to ε > 0.

The main result of the present paper is the following

Theorem 1.

For every ε > 0, γ > 0, and $g \in \mathcal {G}$ we have

$$ {\Delta}_{n} \le C_{\textsc{e}}(\varepsilon, \gamma)\cdot L_{\textsc{e},n}(g,\varepsilon, \gamma), $$

(2.3)

$$ {\Delta}_{n} \le C_{\textsc{r}}(\varepsilon, \gamma)\cdot L_{\textsc{r},n}(g,\varepsilon, \gamma), $$

(2.4)

where

$$ \begin{array}{rcll} C_{\textsc{e}}(\varepsilon, \cdot )&=&A_{\textsc{e}}(\varepsilon, \cdot ),\quad\ \varepsilon\in(0,1],&\text{ in particular, } C_{\textsc{e}}(+0, \cdot )= \infty, \\ C_{\textsc{e}}(\varepsilon, \cdot )&\le&A_{\textsc{e}}(1, \cdot ),\quad\ \varepsilon>1; \\ C_{\textsc{r}}(\varepsilon, \cdot )&=&A_{\textsc{r}}(\varepsilon, \cdot ),\quad\ \varepsilon\in(0,1],&\text{ in particular, } C_{\textsc{r}}(+0, \cdot )= \infty, \\ C_{\textsc{r}}(\varepsilon, \cdot )&\le&\varepsilon A_{\textsc{r}}(\varepsilon, \cdot ),\quad \varepsilon>1, &C_{\textsc{r}}(\infty, \cdot ) = \infty, \end{array} $$

and A_e, A_r are as in Eqs. 1.10 and 1.11.

Observe that inequalities (2.3) and (2.4) with γ = 1 reduce to Eqs. 1.14, 1.15 with C_e(⋅) = C_e(⋅,1), C_r(⋅) = C_r(⋅,1), and with g = g_∗, ε ∈ (0,1] to Eqs. 1.10 and 1.11, respectively. Moreover, Eq. 2.3 also improves Wang–Ahmad inequality (1.9) due to moving the sum ${\sum }_{k=1}^{n}$ inside the modulus sign and under the least upper bound $\sup _{z>0}$ with the range becoming bounded to z < εB_n, so that $A_{6}\le C_{\textsc {e}}(\infty ,1)$. We call inequalities (2.3) and (2.4) analogues of Esseen–Wang–Ahmad’s and Rozovskii’s inequalities.

Since the special cases (1.12) and (1.13) of inequalities (2.3) and (2.4) are natural convergence rate estimates in the Lindeberg–Feller theorem, so are inequalities (2.3) and (2.4) (the latest one under the additional symmetry condition formulated above).

The next statement summarizes all what was said above on the constants A_k, k = 1,…,6.

Corollary 1.

We have

$$ \begin{array}{*{20}l} 0.5409<A_{1}=A_{2}\le1.87, \\ 0.5409<A_{2}\le A_{4}\le A_{\textsc{e}}(1,1)\le2.73, \\ 0.5409<A_{1}\le A_{6}\le C_{\textsc{e}}(\infty,1)\le A_{\textsc{e}}(1,1)\le2.73, \\ A_{3}\le \min\{A_{4},A_{6},A_{\textsc{e}}(\infty,1)\}\le 2.66, \\ A_{5}= A_{\textsc{r}}(1,1)\le 2.73, \end{array} $$

with equalities A₄ = A_e(1,1), $A_{6}=C_{\textsc {e}}(\infty ,1),$ $A_{3}=A_{\textsc {e}}(\infty ,1),$ in the i.i.d. case.

In Gabdullin et al. (2020) we complete corollary 1 by showing, in particular, that

$$ A_{3}>0.3703,\quad A_{5}>0.5685. $$

It is easy to see that the both fractions L_{∙, n} ∈{L_{e, n}, L_{r, n}} are invariant with respect to scale transformations of a function $g\in \mathcal {G}$:

$$ L_{\bullet , n}(cg, \cdot , \cdot ) = L_{\bullet , n}(g, \cdot , \cdot ), \quad c>0. $$

(2.5)

Moreover, extremal properties of the functions

$$ g_{0}(z):= B_{n}g_{0}(z,B_{n})=\min\{z, B_{n}\}, \quad g_{1}(z):= B_{n}g_{1}(z,B_{n})=\max\{z, B_{n}\},\quad z>0, $$

in Eq. 1.1 with a := B_n yield

$$ L_{\bullet , n}(g_{0}, \cdot , \cdot ) \le L_{\bullet , n}(g, \cdot , \cdot ) \le L_{\bullet , n}(g_{1}, \cdot , \cdot ), $$

(2.6)

hence the “universal” upper bounds for the appearing constants C_e and C_r in Eqs. 2.3 and 2.4 are attained at g = g₀. It is also obvious that with the extremal g the both fractions L_{∙, n} ∈{L_{e, n}, L_{r, n}} satisfy

$$ L_{\bullet , n}(g_{0},\varepsilon, \cdot ) = L_{\bullet , n}(g_{*},\varepsilon, \cdot ) \quad\text{for } \varepsilon\le1, $$

(2.7)

$$ L_{\bullet , n}(g_{1},\varepsilon, \cdot ) = L_{\bullet , n}(g_{\textsc{c}},\varepsilon, \cdot ) \quad\text{for } \varepsilon\le1, $$

(2.8)

where, as before,

$$ g_{*}(z)=z,\quad g_{\textsc{c}}(z)\equiv1,\quad z>0. $$

We also note that our proof of inequality (2.3) is completely different from the one by Wang and Ahmad (2016) who used a direct method based on a smoothing inequality and estimates for characteristic functions, similarly to the proof of Esseen’s inequality (1.6) in Esseen (1969). Our proof of Eq. 2.3 is based on estimate (1.10) and property (i) of the class $\mathcal {G}$ (see Eq. 1.1) yielding inequality (2.6) which makes our proof much simpler and shorter.

The next statement establishes some interesting properties and alternative expressions for the introduced fractions L_{e, n}(g, ε, γ) and L_{r, n}(g, ε, γ).

Theorem 2.

For all ε > 0 and γ > 0 we have

$$ L_{\textsc{r},n}(g_{0},\varepsilon, \gamma) = \frac{\gamma}{\varepsilon\vee1} \left|M_{n}(\varepsilon)\right| + \sup\limits_{0 < z < \varepsilon\wedge1} z L_{n}(z), $$

(2.9)

$$ 1 \le L_{\textsc{e},n}(g_{1},\varepsilon, \gamma) \le \max\{\varepsilon, 1\}\cdot \max\{\gamma, 1\} , $$

(2.10)

$$ 1 \le L_{\textsc{r},n}(g_{1},\varepsilon, \gamma) \le \max\{\varepsilon, 1\} \cdot (\gamma + 1), $$

(2.11)

in particular,

$$ L_{\textsc{e},n}(g_{1},\varepsilon,\gamma)\equiv1\quad\text{for }\gamma\le1 \text{ and } \varepsilon\le1, $$

and, in the symmetric case, also

$$ L_{\textsc{e},n}(g_{1},\varepsilon, \gamma)\equiv L_{\textsc{r},n}(g_{1},\varepsilon, \gamma)\equiv1\quad\text{for all } \gamma>0 \text{ and } \varepsilon\le1. $$

3 Proofs

The proof of theorem 1 uses Eq. 2.9, so, we start with the proof of theorem 2.

Proof of theorem 2.

Recall that $g_{0}(z)=\min \limits \{z,B_{n}\}$, $g_{1}(z)=\max \limits \{z,B_{n}\},$ z > 0,

$$ L_{\textsc{e},n}(g,\varepsilon, \gamma) = \sup\limits_{0 < z < \varepsilon} \frac{g(zB_{n})}{zg(B_{n})} \left( \gamma\left|M_{n}(z)\right| + zL_{n}(z) \right), $$

$$ L_{\textsc{r},n}(g,\varepsilon, \gamma) = \gamma \frac{g(\varepsilon B_{n})}{\varepsilon g(B_{n})} \left|M_{n}(\varepsilon)\right| + \sup\limits_{0 < z < \varepsilon}\frac{g(zB_{n})}{g(B_{n})} L_{n}(z), $$

$$ M_{n}(z)=\frac1{{B_{n}^{3}}}\sum\limits_{k=1}^{n} \mu_{k}(zB_{n}) = \frac1{{B_{n}^{3}}}\sum\limits_{k=1}^{n} {\textsf{E}} {X_{k}^{3}}{\bf1}(|X_{k}|<zB_{n}), $$

$$ L_{n}(z) = \frac{1}{{B_{n}^{2}}} \sum\limits_{k=1}^{n} {\sigma_{k}^{2}}(zB_{n}) = \frac{1}{{B_{n}^{2}}} \sum\limits_{k=1}^{n} {\textsf{E}} {X_{k}^{2}}{\bf1} (\left|X_{k}\right| \ge zB_{n}). $$

Representation (2.9) is trivial for ε ≤ 1 due to the observation that g₀(z) = z for z ≤ B_n. As for ε > 1, we have

$$ L_{\textsc{r},n}(g_{0},\varepsilon, \gamma) = \frac{\gamma}{\varepsilon}\left|M_{n}(\varepsilon)\right| + \max\Big\{ \sup\limits_{0 < z < 1} zL_{n}(z), \sup\limits_{1\le z<\varepsilon}L_{n}(z)\Big\} $$

for all γ > 0. Since L_n(z) is left-continuous and non-increasing, we have

$$ \sup\limits_{1\le z<\varepsilon}L_{n}(z)=L_{n}(1)\le\sup\limits_{0 < z < 1} zL_{n}(z), $$

and hence,

$$ L_{\textsc{r},n}(g_{0},\varepsilon, \gamma) =\frac{\gamma}{\varepsilon}\left|M_{n}(\varepsilon)\right| + \sup\limits_{0<z<1}zL_{n}(z), $$

which coincides with the right-hand side of Eq. 2.9 for ε ≥ 1.

To prove (2.10), first, observe that Eq. 1.5 yields

$$ \left|M_{n}(\varepsilon)\right|\le \frac{1}{{B_{n}^{3}}} \sum\limits_{k=1}^{n} {\textsf{E}}|X_{k}|^{3}{\bf1}(|X_{k}|<\varepsilon B_{n})={\Lambda}_{n}(\varepsilon)\le\varepsilon,\quad \varepsilon>0. $$

(3.1)

If ε ≤ 1, then

$$ L_{\textsc{e},n}(g_{1},\varepsilon,\gamma)= \sup\limits_{0 < z < \varepsilon} \left\{ \frac{\gamma}{z}\left|M_{n}(z)\right|+L_{n}(z)\right\}\le \max\{1,\gamma\}\sup\limits_{0 < z < \varepsilon} \left\{ \frac{|{M_{n}(z)}|}{z}+L_{n}(z)\right\} $$

$$ \le \frac{\max\{1, \gamma\}}{{B_{n}^{2}}}\sup\limits_{0 < z < \varepsilon} \left\{ \frac{1}{zB_{n}} \sum\limits_{k=1}^{n} {\textsf{E}}|X_{k}|^{3}{\bf1}(|X_{k}|<z B_{n}) + \sum\limits_{k = 1}^{n} {\textsf{E}} {X_{k}^{2}}{\bf1}(|X_{k}|\ge z B_{n}) \right\} $$

$$ \le \max\{\gamma, 1\} = \max\{\varepsilon, 1\} \cdot \max\{\gamma, 1 \},\quad \gamma>0. $$

If ε > 1, then

$$ L_{\textsc{e},n}(g_{1},\varepsilon, \gamma) = \max \left\{ L_{\textsc{e},n}(g_{1},1, \gamma), \sup\limits_{1\le z<\varepsilon} \left\{ \gamma \left|M_{n}(z)\right| + zL_{n}(z)\right\} \right\},\quad\gamma>0. $$

As we have seen, $L_{\textsc {e},n}(g_{1},1, \gamma )\le \max \limits \{\gamma , 1\}.$ And the second argument of the maximum here can be bounded from above as

$$ \sup\limits_{1\le z<\varepsilon} \left\{ \gamma \left|M_{n}(z)\right| + zL_{n}(z)\right\} $$

$$ \le \frac{\max\{ \gamma, 1 \}}{{B_{n}^{3}}} \sup\limits_{1\le z < \varepsilon} \left\{\sum\limits_{k=1}^{n} {\textsf{E}}|X_{k}|^{3}{\bf1}(|X_{k}|\le zB_{n}) + zB_{n}\sum\limits_{k=1}^{n}{\sigma_{k}^{2}}(z) \right\} $$

$$ \le \max\{ \gamma, 1 \}\sup\limits_{1\le z<\varepsilon}z= \varepsilon \cdot \max\{\gamma, 1\}, $$

which completes the proof of the upper bound in Eq. 2.10. To prove the lower bound in Eq. 2.10, observe that for every ε > 0 and γ > 0 we have

$$ L_{\textsc{e},n}(g_{1},\varepsilon, \gamma) \ge\lim_{z\to0+}\frac{g_{1}(zB_{n})}{zg_{1}(B_{n})} \left( \gamma\left|M_{n}(z)\right| + zL_{n}(z) \right)= \lim_{z\to0+}\left( \frac{\gamma}z\left|M_{n}(z)\right| + L_{n}(z) \right) $$

$$ \ge\lim_{z\to0+}L_{n}(z)=1. $$

Let us prove (2.11). If ε ≤ 1, then for all γ > 0 we have

$$ L_{\textsc{r},n}(g_{1},\varepsilon, \gamma) = \frac\gamma\varepsilon\left|M_{n}(\varepsilon)\right| + \sup\limits_{0<z<\varepsilon}L_{n}(z) =\frac\gamma\varepsilon\left|M_{n}(\varepsilon)\right| + L_{n}(0)=\frac\gamma\varepsilon\left|M_{n}(\varepsilon)\right| +1, $$

which trivially yields the lower bound L_{r, n}(g₁, ε, γ) ≥ 1 and, with Eq. 3.1, also the upper bound L_{r, n}(g₁, ε, γ) ≤ γ + 1. Combining the two-sided bounds, we obtain (2.11) for ε ≤ 1. If ε > 1, then

$$ L_{\textsc{r},n}(g_{1},\varepsilon, \gamma) = \gamma \left|M_{n}(\varepsilon)\right| + \max\Big\{L_{n}(0), \sup\limits_{1\le z<\varepsilon} z L_{n}(z)\Big\}, $$

whence we have

$$ L_{\textsc{r},n}(g_{1},\varepsilon, \gamma)\ge \max\Big\{L_{n}(0), \sup\limits_{1\le z<\varepsilon} z L_{n}(z)\Big\}\ge L_{n}(0)=1. $$

Furthermore, using

$$ \sup\limits_{1\le z<\varepsilon} z L_{n}(z)\le\varepsilon\sup\limits_{1\le z<\varepsilon}L_{n}(z)=\varepsilon L_{n}(0)=\varepsilon $$

and also Eq. 3.1, we obtain an upper bound

$$ L_{\textsc{r},n}(g_{1},\varepsilon, \gamma)\le \gamma \varepsilon + \max\{1,\varepsilon\}= \varepsilon(\gamma + 1), $$

which proves (2.11) also for ε > 1.

The concluding remarks follow from the observation that M_n(z) ≡ 0 for z ≥ 0 in the symmetric case, and hence, the fractions L_{e, n}(g, ε, γ), L_{r, n}(g, ε, γ) are constant with respect to γ > 0. Letting γ → 0 +, we obtain two-sided bounds

$$ 1\le L_{\bullet , n}(g_{1},\varepsilon,\gamma)\le \max\{\varepsilon,1\},\quad \gamma>0, $$

for both fractions L_{∙, n} ∈{L_{e, n}, L_{r, n}}, whence it follows that L_{e, n}(g₁, ε, γ) = L_{r, n}(g₁, ε, γ) ≡ 1 for ε ≤ 1 and γ > 0. □

Proof 2 (Proof of theorem 1).

Recall that $g_{*}(z) \equiv z\in \mathcal {G}$. Let us fix any $g \in \mathcal {G}, n, F_{1}, F_{2}, \ldots , F_{n}$ and consider two cases.

1) If ε ≤ 1, then, due to Eqs. 2.7 and 2.6, the fractions L_{∙, n} ∈{L_{e, n}, L_{r, n}} in Eqs. 1.10, 1.11, 2.3 and 2.4 satisfy

$$ L_{\bullet , n}(\varepsilon, \cdot ) =L_{\bullet , n}(g_{*},\varepsilon, \cdot ) =L_{\bullet , n}(g_{0},\varepsilon, \cdot ) \le L_{\bullet , n}(g,\varepsilon, \cdot ),\quad g\in\mathcal{G}. $$

Using this inequality in Eqs. 1.10 and 1.11 we obtain

$$ {\Delta}_{n} \leq A_{\textsc{e}}(\varepsilon, \cdot ) L_{\textsc{r},n}(\varepsilon, \cdot )\le A_{\textsc{e}}(\varepsilon, \cdot ) L_{\textsc{r},n}(g,\varepsilon, \cdot ), $$

$$ {\Delta}_{n} \le A_{\textsc{r}}(\varepsilon, \cdot ) \cdot L_{\textsc{e},n}(\varepsilon, \cdot )\le A_{\textsc{r}}(\varepsilon, \cdot ) \cdot L_{\textsc{e},n}(g,\varepsilon, \cdot ) $$

for all $g\in \mathcal {G}$, which yields (2.3) and (2.4) with C_e(ε,⋅) ≤ A_e(ε,⋅) and C_r(ε,⋅) ≤ A_r(ε,⋅). Observing that $g_{0}\in \mathcal {G}$ we conclude that inequalities are identities, in fact.

2) Let ε > 1. Since L_{e, n}(g, ε, γ) is non-decreasing with respect to ε, we have

$$ {\Delta}_{n} \le A_{\textsc{e}}(1, \cdot ) L_{\textsc{e},n}(g, 1, \cdot ) \le A_{\textsc{e}}(1, \cdot ) L_{\textsc{e},n}(g, \varepsilon, \cdot ) , $$

for all $g\in \mathcal {G}$, which yields (2.3) with C_e(ε,⋅) ≤ A_e(1,⋅) for all ε > 1. Furthermore, using Eq. 2.9 we obtain

$$ L_{\textsc{r},n}(g_{0},\varepsilon, \gamma) = \frac{\gamma}{\varepsilon} \left|M_{n}(\varepsilon)\right| + \sup\limits_{0<z<1} z L_{n}(z). $$

Due to the monotonicity of L_n(z), the second term here can be bounded from below as

$$ \sup\limits_{0<z<1}zL_{n}(z)=\sup\limits_{0<z<\varepsilon}\frac{z}{\varepsilon}L_{n}\left( \frac{z}\varepsilon\right)\ge \frac1\varepsilon\sup\limits_{0<z<\varepsilon}zL_{n}(z), $$

hence, we obtain a lower bound

$$ L_{\textsc{r},n}(g_{0},\varepsilon, \gamma) \ge \frac{1}{\varepsilon} \left (\gamma\left|M_{n}(\varepsilon)\right| + \sup\limits_{0<z<\varepsilon} z L_{n}(z)\right) =\frac1\varepsilon L_{\textsc{r},n}(g_{*},\varepsilon,\gamma). $$

Now observing that

$$ L_{\textsc{r},n}(\varepsilon, \cdot ) =L_{\textsc{r},n}(g_{*},\varepsilon, \cdot ) \le \varepsilon L_{\textsc{r},n}(g_{0},\varepsilon, \cdot ) \le \varepsilon L_{\textsc{r},n}(g,\varepsilon, \cdot ),\quad g\in\mathcal{G}, $$

and using Eq. 1.11, we obtain (2.4) with C_r(ε,⋅) ≤ εA_r(ε,⋅) for ε ≥ 1.

The fact that $C_{\textsc {e}}(+0, \cdot ) = C_{\textsc {r}}(+0, \cdot ) = +\infty$ for all γ > 0 trivially follows either from unboundedness of A_e(+ 0,⋅) and A_r(+ 0,⋅), or, directly, from Eq. 2.6 and

$$ \lim_{\varepsilon\to+0}L_{\textsc{e},n}(g_{0},\varepsilon,\gamma) = \lim_{\varepsilon\to+0}L_{\textsc{r},n}(g_{0},\varepsilon,\gamma)=0, \quad \gamma > 0. $$

To prove that $\lim \limits _{\varepsilon \to \infty } C_{\textsc {r}}(\varepsilon , \cdot ) = +\infty$, assume that the random summands X₁,…, X_n are i.i.d. and have finite third-order moments. Then, due to Eq. 2.9, we have

$$ \lim\limits_{\varepsilon \to\infty}L_{\textsc{r},n}(g_{0},\varepsilon, \gamma) = \lim\limits_{\varepsilon \to\infty}\frac{\gamma}{\varepsilon} \left|M_{n}(\varepsilon)\right| + \sup\limits_{0<z<1} zL_{n}(z) =\sup\limits_{0<z<1} zL_{n}(z). $$

On the other hand, in Gabdullin et al. (2019a, Theorem 3) it is shown that

$$ \sup\limits_{F} \limsup\limits_{n \rightarrow +\infty} \frac{{\Delta}_{n}}{\sup\limits_{0<z<1} zL_{n}(z)} = +\infty, $$

where the least upper bound is taken over all identical distribution functions F₁ = … = F_n = F of the random summands X₁,…, X_n with finite third-order moments. Since estimate (2.4) must hold also in this particular case, we should necessarily have ${C_{\textsc {r}}(\varepsilon , \cdot )\to \infty }$ as ${\varepsilon \to \infty }$. □