Complete Convergence for Maximal Sums of Negatively Associated Random Variables

Abstract

Necessary and sufficient conditions are given for the complete convergence of maximal sums of identically distributed negatively associated random variables. The conditions are expressed in terms of integrability of random variables. Proofs are based on new maximal inequalities for sums of bounded negatively associated random variables.

1. Introduction

The paper by Hsu and Robbins [1] initiated a great interest to the complete convergence of sums of independent random variables. Their research was continued by Erdös [2, 3], Spitzer [4], and Baum and Katz [5]. Kruglov et al. [6] proved two general theorems that provide sufficient conditions for the complete convergence for sums of arrays of row-wise independent random variables. In the paper of Kruglov and Volodin [7], a criterion was proved for the complete convergence of sums of independent identically distributed random variable in a rather general setting. Taylor et al. [8] and Chen et al. [9, 10] demonstrated that many known sufficient conditions for complete convergence of sums of independent random variables can be transformed to sufficient conditions for the complete convergence of sums of negatively associated random variables. Here we give necessary and sufficient conditions for the complete convergence of maximal sums of negatively associated identically distributed random variables. They resemble the criterions presented by Baum and Katz [5] and by Kruglov and Volodin [7] for the complete convergence of sums of independent identically distributed random variables. Theorems 2.3 and 2.5 are our main results. Theorem 2.3 is new even for independent random variables.

In what follows we assume that all random variables under consideration are defined on a probability space (Ω,ℱ,𝑃). We use standard notations, in particular, 𝐼_𝐴 denotes the indicator function of a set 𝐴⊆Ω. Recall the notion of negatively associated random variables and some properties of such random variables.

Definition 1.1. Random variables 𝑋₁,…,𝑋_𝑛 are called negatively associated if Cov(𝑓(𝑋_𝑖1,…,𝑋_𝑖𝑘),𝑔(𝑋_𝑗1,…,𝑋_𝑗𝑚))≤0 (1.1) for any pair of nonempty disjoint subsets 𝐴={𝑖₁,…,𝑖_𝑘} and 𝐵={𝑗₁,…,𝑗_𝑚},𝑘+𝑚≤𝑛, of the set {1,…,𝑛} and for any bounded coordinate-wise increasing real functions 𝑓(𝑥_𝑖1,…,𝑥_𝑖𝑘) and 𝑔(𝑥_𝑗1,…,𝑥_𝑗𝑚),𝑥₁,…,𝑥_𝑛∈ℝ=(−∞,∞). Random variables 𝑋_𝑛,𝑛∈ℕ={1,2,…,} are negatively associated if for any 𝑛∈ℕ random variables 𝑋₁,…,𝑋_𝑛 are negatively associated.
In this definition the coordinate-wise increasing functions 𝑓 and 𝑔 may be replaced by coordinate-wise decreasing functions. Indeed, if 𝑓 and 𝑔 are coordinate-wise decreasing functions, then −𝑓 and −𝑔 are coordinate-wise increasing functions and the covariance (1.1) coincides with the covariance for −𝑓 and −𝑔.

Theorem A. Let 𝑋_𝑛,𝑛∈ℕ, be negatively associated random variables. Then for every 𝑎_𝑛,𝑏_𝑛∈ℝ,𝑎_𝑛≤𝑏_𝑛, the random variables 𝑌_𝑛=𝑎_𝑛𝐼_{(−∞,𝑎𝑛)}(𝑋_𝑛)+𝑋_𝑛𝐼_{[𝑎𝑛,𝑏𝑛]}(𝑋_𝑛)+𝑏_𝑛𝐼_{(𝑏𝑛,∞)}(𝑋_𝑛),𝑛∈ℕ, are negatively associated. For every 𝑛∈ℕ and 𝑥₁,…,𝑥_𝑛∈ℝ, the inequalities 𝑃{𝑋₁≤𝑥₁,…,𝑋_𝑛≤𝑥_𝑛}≤^𝑛_𝑘=1𝑃{𝑋_𝑘≤𝑥_𝑘},𝑃{𝑋₁≥𝑥₁,…,𝑋_𝑛≥𝑥_𝑛}≤^𝑛_𝑘=1𝑃{𝑋_𝑘≥𝑥_𝑘} (1.2) hold.

Proof. It can be found in Taylor et al. [8].

Theorem B. Let 𝑋_𝑛 be negatively associated random variables. Let 𝑋^∗_𝑛,𝑛∈ℕ, be independent random variables such that 𝑋_𝑘 and 𝑋^∗_𝑘 are identically distributed for every 𝑘=1,…,𝑛. Then 𝐸exp{𝑋₁+⋯+𝑋_𝑛}≤𝐸exp{𝑋^∗₁+⋯+𝑋^∗_𝑛}. (1.3) If 𝐸|𝑋_𝑘|^𝑝<∞ and 𝐸𝑋_𝑘=0 for all 𝑘=1,…,𝑛 and for some 𝑝≥1, then 𝐸|max^𝑟_{1≤𝑟≤𝑛𝑘=1}𝑋_𝑘|^𝑝≤2𝐸|^𝑛_𝑘=1𝑋^∗_𝑘|^𝑝,𝑛∈ℕ. (1.4)

Proof. It can be found in Qi-Man Shao [11].

2. Main Results

Our basic theorems will be stated in terms of special functions. They were introduced in Kruglov and Volodin [7].

Definition 2.1. A nonnegative function ℎ(𝑥),𝑥∈[0,∞), belongs to the class ℋ_𝑞 for some 𝑞∈(0,2) if it is nondecreasing, is not equal to zero identically, and limsup_𝑥→∞ℎ(2𝑥)ℎ(𝑥)<∞,limsup_𝑦→∞𝑦^2/𝑞−1ℎ(𝑦)^∞_𝑦ℎ(𝑥)𝑥^−2/𝑞𝑑𝑥<∞. (2.1)(2.2)
The class ℋ_𝑞 contains all nondecreasing nonnegative functions slowly varying at infinity which are not equal to zero identically, and in particular, ln^𝛽(1+𝑥) with 𝛽>0. The functions 𝑥^𝛼 and 𝑥^𝛼ln^𝛽(1+𝑥) with 𝛼∈[0,2/𝑞−1) and 𝛽>0 are also in ℋ_𝑞.

Remark 2.2. If a nonnegative function ℎ(𝑥),𝑥∈[0,∞), is nondecreasing and satisfies condition (2.1), then ℎ(𝑥)≤Δ𝑥^𝑙 (2.3) for all 𝑥 greater than some 𝑥₀≥1 and for some Δ>0 and 𝑙>0.

Proof. We may assume that ℎ(𝑥)>0 for all 𝑥 greater than some 𝑥₀≥1. From condition (2.1), it follows that sup_𝑥≥𝑥0ℎ(2𝑥)/ℎ(𝑥)=𝑑<∞. Choose a number 𝑙>0 such that 𝑑≤2^𝑙. If 𝑥≥2𝑥₀, then 2^𝑛𝑥₀≤𝑥<2^𝑛+1𝑥₀ for some 𝑛∈ℕ and ℎ(𝑥)≤ℎ(2^𝑛+1𝑥₀)≤𝑑^𝑛+1ℎ(𝑥₀)≤2^𝑙(𝑛+1)ℎ(𝑥₀)≤2^𝑙𝑥^𝑙ℎ(𝑥₀). Inequality (2.3) holds for all 𝑥≥𝑥₀ with Δ=2^𝑙ℎ(𝑥₀).

Theorem 2.3. Let 𝑋_𝑛,𝑛∈ℕ, be negatively associated identically distributed random variables, 𝑆_𝑛=𝑋₁+⋯+𝑋_𝑛,0<𝑞<2,𝑟>1. Let ℎ(𝑥),𝑥∈[0,∞), be a function which is nondecreasing, is not equal to zero identically, and satisfies condition (2.1). Then the following conditions are equivalent: 𝐸(|𝑋₁|^𝑟𝑞ℎ(|𝑋₁|^𝑞))<∞,𝐸𝑋₁=0,for𝑞≥1,_∞𝑛=1𝑛^𝑟−2ℎ(𝑛)𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀𝑛^1/𝑞}<∞,∀𝜀>0,_∞𝑛=1𝑛^𝑟−2ℎ(𝑛)𝑃{sup_𝑘≥𝑛|𝑘^−1/𝑞𝑆_𝑘|>𝜀}<∞,∀𝜀>0. (2.4)(2.5)(2.6)

Corollary 2.4. Let 𝑋_𝑛,𝑛∈ℕ, be negatively associated identically distributed random variables, 𝑆_𝑛=𝑋₁+⋯+𝑋_𝑛,0<𝑞<2,𝑟>1,𝛽≥0. The following conditions are equivalent: 𝐸(|𝑋₁|^𝑟𝑞(ln(1+|𝑋₁|))^𝛽𝑞)<∞,𝐸𝑋₁=0,for𝑞≥1,_∞𝑛=1𝑛^𝑟−2(ln𝑛)^𝛽𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀𝑛^1/𝑞}<∞,∀𝜀>0,_∞𝑛=1𝑛^𝑟−2(ln𝑛)^𝛽𝑃{sup_𝑘≥𝑛|𝑘^−1/𝑞𝑆_𝑘|>𝜀}<∞,∀𝜀>0. (2.7)

A part of Theorem 2.3 can be generalized to a larger range of 𝑟,𝑟≥1, under additional restrictions on functions ℎ(𝑥),𝑥∈[0,∞).

Theorem 2.5. Let 𝑋_𝑛,𝑛∈ℕ, be negatively associated identically distributed random variables, 𝑆_𝑛=𝑋₁+⋯+𝑋_𝑛,ℎ∈ℋ_𝑞,0<𝑞<2,𝑟≥1. Then the following conditions are equivalent: 𝐸(|𝑋₁|^𝑟𝑞ℎ(|𝑋₁|^𝑞))<∞,𝐸𝑋₁=0,for𝑞≥1,_∞𝑛=1𝑛^𝑟−2ℎ(𝑛)𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀𝑛^1/𝑞}<∞,∀𝜀>0. (2.8)(2.9)

Corollary 2.6. Let 𝑋_𝑛,𝑛∈ℕ, be negatively associated identically distributed random variables, 𝑆_𝑛=𝑋₁+⋯+𝑋_𝑛,0<𝑞<2,𝛽≥0. The following conditions are equivalent: 𝐸(|𝑋₁|^𝑞(ln(1+|𝑋₁|))^𝛽𝑞)<∞,𝐸𝑋₁=0,for𝑞≥1,_∞𝑛=1(ln𝑛)^𝛽𝑛𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀𝑛^1/𝑞}<∞,∀𝜀>0. (2.10)

Proof of Theorem 2.3. The theorem is obvious if the random variable 𝑋₁ is degenerate, that is, 𝑃{𝑋₁=const.}=1. From now on we suppose that the random variable 𝑋₁ is not degenerate. Denote 𝛿=𝑟−1𝑟+𝑙if0<𝑟𝑞<1,𝑟−14(𝑟+𝑙)if1≤𝑟𝑞≤2,2−𝑞4𝑞(𝑟+𝑙)if𝑟𝑞>2, (2.11) where 𝑙 is the same as in (2.3). Define the function 𝑓(𝑥),𝑥∈ℝ,𝑓(𝑥)=𝜀𝑛^1/𝑞𝛿if𝑥>𝜀𝑛^1/𝑞𝛿,𝑥if|𝑥|≤𝜀𝑛^1/𝑞𝛿,−𝜀𝑛^1/𝑞𝛿if𝑥<−𝜀𝑛^1/𝑞𝛿, (2.12) where 𝜀>0 is a fixed number. By Theorem A the random variables 𝑌_𝑘=𝑓(𝑋_𝑘),𝑘=1,…,𝑛, are negatively associated. Put 𝑆^′_𝑘=𝑌₁+⋯+𝑌_𝑘,𝑘=1,…,𝑛. Note that |𝑌₁|≤|𝑋₁| and 𝑌₁→𝑋₁ as 𝑛→∞. If 𝐸|𝑋₁|^𝑟𝑞<∞, then by the dominated convergence theorem we have lim_𝑛→∞𝐸|𝑌₁|^𝑟𝑞=𝐸|𝑋₁|^𝑟𝑞,lim_𝑛→∞𝐸|𝑌₁−𝐸𝑌₁|^𝑟𝑞=𝐸|𝑋₁−𝐸𝑋₁|^𝑟𝑞if𝑟𝑞≥1. (2.13) Prove that (2.4)⇒(2.5). Assume that 0<𝑟𝑞<1. The probability 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀𝑛^1/𝑞} can be estimated as follows: 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀𝑛^1/𝑞}=𝑃{{max_{1≤𝑘≤𝑛}|𝑆^′_𝑘|>𝜀𝑛^1/𝑞}∩^𝑛_𝑘=1{𝑋_𝑘=𝑌_𝑘}}+𝑃{{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀𝑛^1/𝑞}∩(^𝑛_𝑘=1{𝑋_𝑘≠𝑌_𝑘})}≤𝑃{max_{1≤𝑘≤𝑛}|𝑆^′_𝑘|>𝜀𝑛^1/𝑞}+𝑃{max_{1≤𝑘≤𝑛}|𝑋_𝑘|>𝜀𝑛^1/𝑞𝛿}. (2.14) We intend to use Lemma 3.1 from the third part of the paper. Put 𝛾=𝑟𝑞,𝑥=𝜀𝑛^1/𝑞,𝑐=𝜀𝑛^1/𝑞(𝑟−1)/(𝑟+𝑙). From (2.13) it follows that the inequality 𝐸|𝑌₁|^𝛾<2𝐸|𝑋₁|^𝛾 holds for all 𝑛∈ℕ greater than some 𝑛₀. By inequality (3.1), we get, for all 𝑛>𝑛₀,𝑃{max_{1≤𝑘≤𝑛}|𝑆^′_𝑘|>𝜀𝑛^1/𝑞𝑥}≤2exp{𝑐−𝑥𝑐ln(𝑥𝑐^𝛾−1𝑛𝐸|𝑌₁|^𝛾+1)}≤2exp{𝑟+𝑙−𝑟−1𝑟+𝑙𝜀𝑟−1ln(^𝑟q(𝑟−1)^𝑟𝑞−12𝐸|𝑋₁|^𝑟𝑞(𝑟+𝑙)^𝑟𝑞−1𝑛^𝑟−1+1)}≤2exp{𝑟+𝑙𝜀𝑟−1}(^𝑟𝑞(𝑟−1)^𝑟𝑞−12𝐸|𝑋₁|^𝑟𝑞(𝑟+𝑙)^𝑟𝑞−1)^{−(𝑟+𝑙)/(𝑟−1)}𝑛^{−(𝑟+𝑙)}=𝐶𝑛^{−(𝑟+𝑙)}. (2.15) These inequalities and (2.3) imply that ^∞_𝑛=𝑛0+1𝑛^𝑟−2ℎ(𝑛)𝑃{max_{1≤𝑘≤𝑛}|𝑆^′_𝑘|>𝜀𝑛^1/𝑞}≤Δ𝐶^∞_𝑛=𝑛0+1𝑛^{𝑟+𝑙−2}𝑛^{−(𝑟+𝑙)}<∞. (2.16) Since 𝑃{max_{1≤𝑘≤𝑛}|𝑋_𝑘|>𝜀𝑛^1/𝑞𝛿}≤𝑛𝑃{|𝑋₁|>𝜀𝑛^1/𝑞𝛿}, we obtain ^∞_𝑛=𝑛0+1𝑛^𝑟−2ℎ(𝑛)𝑃{max_{1≤𝑘≤𝑛}|𝑋_𝑘|>𝜀𝑛^1/𝑞𝛿}≤^∞_𝑛=𝑛0+1𝑛^𝑟−1ℎ(𝑛)𝑃{|𝑋₁|>𝜀𝑛^1/𝑞𝑋𝛿}≤𝐸(|¹|𝜀𝛿^𝑟𝑞𝑋ℎ(|¹|𝜀𝛿^𝑞)). (2.17) The last inequality holds by Lemma 3.2. From (2.1) it follows that 𝑋𝐸(|¹|𝜀𝛿^𝑟𝑞𝑋ℎ(|¹|𝜀𝛿^𝑞))≤const.𝐸(|𝑋₁|^𝑟𝑞ℎ(|𝑋₁|^𝑞)). (2.18) Condition (2.4) and inequalities (2.14)–(2.17) imply (2.5) for 0<𝑟𝑞<1.
Now assume that 𝑟𝑞≥1. First we consider the case 1≤𝑟𝑞≤2. Note that 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀𝑛^1/𝑞}≤𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘−𝐸𝑆^′_𝑛|>𝜀𝑛^1/𝑞−max_{1≤𝑘≤𝑛}|𝐸𝑆^′_𝑘|}. (2.19)
By Lemma 3.4, we have max_{1≤𝑘≤𝑛}𝐸|𝑆^′_𝑘|𝑛^−1/𝑞→0 as 𝑛→∞, and hence 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀𝑛^1/𝑞}≤𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘−𝐸𝑆^′_𝑛|>𝜀𝑛^1/𝑞2} (2.20) for all 𝑛∈ℕ greater than some 𝑛^′₀. The probability on the right-hand side can be estimated as follows: 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘−𝐸𝑆^′_𝑘|>𝜀𝑛^1/𝑞2}=𝑃{{max_{1≤𝑘≤𝑛}|𝑆^′_𝑘−𝐸𝑆^′_𝑘|>𝜀𝑛^1/𝑞2}∩^𝑛_𝑘=1{𝑋_𝑘=𝑌_𝑘}}+𝑃{{max_{1≤𝑘≤𝑛}|𝑆_𝑘−𝐸𝑆^′_𝑘|>𝜀𝑛^1/𝑞2}∩(^𝑛_𝑘=1{𝑋_𝑘≠𝑌_𝑘})}≤𝑃{max_{1≤𝑘≤𝑛}|𝑆^′_𝑘−𝐸𝑆^′_𝑘|>𝜀𝑛^1/𝑞2}+𝑃{max_{1≤𝑘≤𝑛}|𝑋_𝑘|>𝜀𝑛^1/𝑞𝛿}. (2.21) In order to apply Lemma 3.1, put 𝛾=𝑟𝑞,𝑥=𝜀𝑛^1/𝑞/2,𝑐=𝜀𝑛^1/𝑞(𝑟−1)/(4(𝑟+𝑙)). From (2.13) it follows that the inequality 𝐸|𝑌₁−𝐸𝑌₁|^𝛾<2𝐸|𝑋₁−𝐸𝑋₁|^𝛾 holds for all 𝑛∈ℕ greater than some 𝑛₀^′′≥𝑛^′₀. By inequality (3.2) we get, for 𝑛>𝑛₀^′′,𝑃{max_{1≤𝑘≤𝑛}|𝑆^′_𝑘−𝐸𝑆^′_𝑘|>𝜀𝑛^1/𝑞2}𝑥≤4exp{2^𝛾−1𝑐−𝑥2𝑐ln(𝑥𝑐^𝛾−1𝑛𝐸|𝑌₁−𝐸𝑌₁|^𝛾+1)}≤4exp{2(𝑟+𝑙)−𝑟−1𝑟+𝑙𝜀𝑟−1ln(^𝑟𝑞(𝑟−1)^𝑟𝑞−12𝐸|𝑋₁−𝐸𝑋₁|^𝑟𝑞(4(𝑟+𝑙))^𝑟𝑞−1𝑛^𝑟−1+1)}≤4exp{2(𝑟+𝑙)𝜀𝑟−1}(^𝑟𝑞(𝑟−1)^𝑟𝑞−12𝐸|𝑋₁−𝐸𝑋₁|^𝑟𝑞(4(𝑟+𝑙))^𝑟𝑞−1)^{−(𝑟+𝑙)/(𝑟−1)}𝑛^{−(𝑟+𝑙)}=𝐶₁𝑛^{−(𝑟+𝑙)}. (2.22) These inequalities and (2.3) imply ^∞_{𝑛=𝑛0′′+1}𝑛^𝑟−2ℎ(𝑛)𝑃{max_{1≤𝑘≤𝑛}|𝑆^′_𝑘−𝐸𝑆^′_𝑘|>𝜀𝑛^1/𝑞2}≤Δ𝐶₁^∞_{𝑛=𝑛0′′+1}𝑛^{𝑟+𝑙−2}𝑛^{−(𝑟+𝑙)}<∞. (2.23) From this inequality and (2.17) with 𝑛₀^′′ instead of 𝑛₀, (2.20) and (2.21), it follows that (2.5) holds for 1≤𝑟𝑞≤2. The case 𝑟𝑞>2 can be considered in the same way. In order to apply Lemma 3.1, put 𝛾=2,𝑥=𝜀𝑛^1/𝑞/2,𝑐=𝜀𝑛^1/𝑞(2−𝑞)/(4𝑞(𝑟+𝑙)). From (2.13) it follows that the inequality 𝐸|𝑌₁−𝐸𝑌₁|²<2𝐸|𝑋₁−𝐸𝑋₁|² holds for all 𝑛∈ℕ greater than some 𝑛₀^′′′≥𝑛^′₀. By inequality (3.2), we get, for 𝑛>𝑛₀^′′′,𝑃{max_{1≤𝑘≤𝑛}|𝑆^′_𝑘−𝐸𝑆^′_𝑘|>𝜀𝑛^1/𝑞2}𝑥≤4exp{−𝑥2𝑐2𝑐ln(x𝑐𝑛𝐸|𝑌₁−𝐸𝑌₁|²+1)}≤4exp{𝑞(𝑟+𝑙)−2−𝑞𝑞(𝑟+𝑙)𝜀2−𝑞ln(²(2−𝑞)16𝐸|𝑋₁−𝐸𝑋₁|²𝑛𝑞(𝑟+𝑙)^2/𝑞−1+1)}≤4exp{𝑞(𝑟+𝑙)𝜀2−𝑞}(²(2−𝑞)16𝐸|𝑋₁−𝐸𝑋₁|²)𝑞(𝑟+𝑙)^{−𝑞(𝑟+𝑙)/(2−𝑞)}𝑛^{−(𝑟+𝑙)}=𝐶₂𝑛^{−(𝑟+𝑙)}. (2.24) These inequalities and (2.3) imply ^∞_{𝑛=𝑛0′′′+1}𝑛^𝑟−2ℎ(𝑛)𝑃{max_{1≤𝑘≤𝑛}|𝑆^′_𝑘−𝐸𝑆^′_𝑘|>𝜀𝑛^1/𝑞2}≤Δ𝐶₂^∞_{𝑛=𝑛0′′′+1}𝑛^{𝑟+𝑙−2}𝑛^{−(𝑟+𝑙)}<∞. (2.25) From this inequality and (2.17) with 𝑛₀^′′′ instead of 𝑛₀, (2.20) and (2.21), it follows that (2.5) holds for 𝑟𝑞>2.
Prove that (2.5)⇒(2.6). Note that^∞_𝑛=3𝑛^𝑟−2ℎ(𝑛)𝑃{sup_𝑘≥𝑛|𝑘^−1/𝑞𝑆_𝑘|>𝜀}=^∞2_𝑗=1^𝑗+1_𝑛=2𝑗+1𝑛^𝑟−2ℎ(𝑛)𝑃{sup_𝑘≥𝑛|𝑘^−1/𝑞𝑆_𝑘≤|>𝜀}_∞𝑗=12^{(𝑗+1)(𝑟−1)}ℎ(2^𝑗+1)𝑃{sup_𝑘≥2𝑗|𝑘^−1/𝑞𝑆_𝑘≤|>𝜀}_∞𝑗=12^{(𝑗+1)(𝑟−1)}ℎ(2^𝑗+1)^∞_𝑖=𝑗𝑃{max_{2𝑖≤𝑘≤2𝑖+1}|𝑘^−1/𝑞𝑆_𝑘=|>𝜀}_{∞𝑖𝑖=1𝑗=1}2^{(𝑗+1)(𝑟−1)}ℎ(2^𝑗+1)𝑃{max_{2𝑖≤𝑘≤2𝑖+1}|𝑘^−1/𝑞𝑆_𝑘≤2|>𝜀}_𝑟−12^𝑟−1−1^∞_𝑖=12^{(𝑖+1)(𝑟−1)}ℎ(2^𝑖+1)𝑃{max_{2𝑖≤𝑘≤2𝑖+1}|𝑆_𝑘|>𝜀2^i/𝑞}. (2.26) The last series can be estimated as follows: ^∞_𝑖=12^{(𝑖+1)(𝑟−1)}ℎ(2^𝑖+1)𝑃{max_{2𝑖≤𝑘≤2𝑖+1}|𝑆_𝑘|>𝜀2^i/𝑞}≤2^5𝑟−3ℎ(2⁵)+max{1,2^2−𝑟}^∞2_𝑖=1^𝑖+2_{𝑛=2𝑖+1+1}𝑛^𝑟−2ℎ(𝑛)𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀2^−2/𝑞𝑛^1/𝑞}≤2^5𝑟−3ℎ(2⁵)+max{1,2^2−𝑟}^∞_𝑛=1𝑛^𝑟−2ℎ(𝑛)𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀2^−2/𝑞𝑛^1/𝑞}. (2.27) Condition (2.5) and these inequalities imply (2.6).
Prove that (2.6)⇒(2.4). The sequence {𝑃{sup_𝑘≥𝑛|𝑘^−1/𝑞𝑆_𝑘|>𝜀}}_𝑛≥1 decreases to zero for any 𝜀>0. Indeed, if the sequence converges to a number 𝑎>0, then ∞=𝑎^∞_𝑛=1𝑛^𝑟−2ℎ(𝑛)≤^∞_𝑛=1𝑛^𝑟−2ℎ(𝑛)𝑃{sup_𝑘≥𝑛|𝑘^−1/𝑞𝑆_𝑘|>𝜀}<∞. (2.28) Note that 𝑃{max_{𝑛≤𝑘≤2𝑛}|𝑆_𝑘|>𝜀(2𝑛)^1/𝑞}≤𝑃{sup_𝑘≥𝑛|𝑘^−1/𝑞𝑆_𝑘|>𝜀}→0 as 𝑛→∞.
With the help of (1.2), we obtain 𝑃{max_{𝑛<𝑘≤2𝑛}𝑋_𝑘>𝑛^1/𝑞}=1−𝑃{max_{𝑛<𝑘≤2𝑛}𝑋_𝑘≤𝑛^1/𝑞}≥1−^2𝑛_𝑘=𝑛+1𝑃{𝑋_𝑘≤𝑛^1/𝑞}=1−(1−𝑃{𝑋₁>𝑛^1/𝑞})^𝑛,𝑃{max_{𝑛<𝑘≤2𝑛}(−𝑋_𝑘)>𝑛^1/𝑞}=1−𝑃{max_{𝑛<𝑘≤2𝑛}(−𝑋_𝑘)≤𝑛^1/𝑞}≥1−^2𝑛_𝑘=𝑛+1𝑃{𝑋_𝑘≥−𝑛^1/𝑞}=1−(1−𝑃{−𝑋₁>𝑛^1/𝑞})^𝑛. (2.29) Denote 𝑎⁺_𝑛=𝑃{𝑋₁>𝑛^1/𝑞} and 𝑎⁻_𝑛=𝑃{−𝑋₁>𝑛^1/𝑞}. Note that 𝑃{|𝑋₁|>𝑛^1/𝑞}=𝑎⁺_𝑛+𝑎⁻_𝑛 and 1−(1−𝑎^±_𝑛)^𝑛≤𝑃{max_{𝑛<𝑘≤2𝑛}|𝑋_𝑘|>𝑛^1/𝑞}≤𝑃{max_{𝑛≤𝑘≤2𝑛}|𝑆_𝑘|>𝑛^1/𝑞/2}. Since 𝑃{max_{𝑛≤𝑘≤2𝑛}|𝑆_𝑘|>𝑛^1/𝑞/2}→0 and 𝑎^±_𝑛≤𝑃{|𝑋₁|>𝑛^1/𝑞}→0 as 𝑛→∞, then 𝑛ln(1−𝑎^±_𝑛)>−ln2 and 0≤𝑎⁺_𝑛+𝑎⁻_𝑛<1 for all 𝑛∈ℕ greater than some 𝑛₀. By the inequalities −ln(1−𝑥)≥𝑥 for 𝑥∈[0,1) and 1−𝑒^𝑥≥𝑒^𝑥|𝑥| for 𝑥≤0, we obtain 𝑛𝑎^±_𝑛≤−𝑛ln(1−𝑎^±_𝑛)≤2(1−𝑒^{𝑛ln(1−𝑎±𝑛)})=2(1−(1−𝑎^±_𝑛)^𝑛)≤2𝑃{max_{1≤𝑘≤𝑛}|𝑋_𝑘|>𝑛^1/𝑞} (2.30) for all 𝑛>𝑛₀, and ^∞_𝑛=𝑛0+1𝑛^𝑟−1ℎ(𝑛)𝑃{|𝑋₁|>𝑛^1/𝑞}=_𝑛=𝑛0+1𝑛^𝑟−1ℎ(𝑛)(𝑎⁺_𝑛+𝑎⁻_𝑛)≤4^∞_𝑛=𝑛0+1𝑛^𝑟−2ℎ(𝑛)𝑃{max_{𝑛<𝑘≤2𝑛}|𝑋_𝑘|>𝑛^1/𝑞}≤4^∞_𝑛=1𝑛^𝑟−2ℎ(𝑛)𝑃{max_{𝑛≤𝑘≤2𝑛}|𝑆_𝑘𝑛|>^1/𝑞2}≤4^∞_𝑛=1𝑛^𝑟−2ℎ(𝑛)𝑃{max_{𝑛≤𝑘≤2𝑛}|𝑘^−1/𝑞𝑆_𝑘2|>^−1/𝑞2}≤4^∞_𝑛=1𝑛^𝑟−2ℎ(𝑛)𝑃{sup_𝑘≥𝑛|𝑘^−1/𝑞𝑆_𝑘2|>^−1/𝑞2}<∞. (2.31) It follows by Lemma 3.3 that 𝐸(|𝑋₁|^𝑟𝑞ℎ(|𝑋₁|^𝑞))<∞.
Now we will prove that 𝑎=𝐸𝑋₁=0 provided that 1≤𝑞<2. Assume that 𝑎≠0. Since 𝑛|𝑎|≤|𝑆_𝑛−𝑛𝑎|+|𝑆_𝑛|, then 1=𝑃{|𝑆_𝑛−𝑛𝑎|+|𝑆_𝑛|>|𝑎|𝑛^1/𝑞2}≤𝑃{|𝑆_𝑛−𝑛𝑎|>|𝑎|𝑛^1/𝑞4}+𝑃{|𝑆_𝑛|>|𝑎|𝑛^1/𝑞4}, (2.32) and hence ∞=^∞_𝑛=1𝑛^𝑟−2ℎ(𝑛)≤^∞_𝑛=1𝑛^𝑟−2ℎ(𝑛)𝑃{|𝑆_𝑛−𝑛𝑎|>|𝑎|𝑛^1/𝑞4}+^∞_𝑛=1𝑛^𝑟−2ℎ(𝑛)𝑃{|𝑆_𝑛|>|𝑎|𝑛^1/𝑞4}. (2.33) But this contradicts the convergence of the series on the right-hand side of the inequality. The equality 𝐸X₁=0 is proved.

Proof of Theorem 2.5. Both theorems overlap. We need to consider only the case 𝑟=1. Prove that (2.8)⇒(2.9). Define random variables 𝑍_𝑛=−𝜀𝑛^1/𝑞𝐼_{(−∞,−𝜀𝑛1/𝑞)}(𝑋_𝑛)+𝑋_𝑛𝐼_{[−𝜀𝑛1/𝑞,𝜀𝑛1/𝑞]}(𝑋_𝑛)+𝜀𝑛^1/𝑞𝐼_{(𝜀𝑛1/𝑞,∞)}(𝑋_𝑛),𝑛∈ℕ. By Theorem A the random variables 𝑍_𝑛,𝑛∈ℕ, as well as 𝑍_𝑛−𝐸𝑍_𝑛,𝑛∈ℕ, are negatively associated. Denote 𝑆_𝑘^′′=(𝑍₁−𝐸𝑍₁)+⋯+(𝑍_𝑘−𝐸𝑍_𝑘). By Theorem B there exist independent random variables 𝑋^∗₁,…,𝑋^∗_𝑛 such that the random variables 𝑍_𝑘−𝐸𝑍_𝑘 and 𝑋^∗_𝑘 are identically distributed for all 𝑘=1,…,𝑛, and 𝐸(max_{1≤𝑘≤𝑛}|𝑆_𝑘^′′|²)≤2𝐸(𝑋^∗₁+⋯+𝑋^∗_𝑛)². Similarly to (2.21), we can prove that 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝜀𝑛^1/𝑞}≤𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘^′′−𝐸𝑆_𝑘^′′|>𝜀𝑛^1/𝑞2}+𝑃{max_{1≤𝑘≤𝑛}|𝑋_𝑘|>𝜀𝑛^1/𝑞} (2.34) for all 𝑛∈ℕ grater than some 𝑛₀. In the same way as (2.17), one can prove that ^∞_𝑛=𝑛0+1ℎ(𝑛)𝑛𝑃{max_{1≤𝑘≤𝑛}|𝑋_𝑘|>𝜀𝑛^1/𝑞}≤^∞_𝑛=𝑛0+1ℎ(𝑛)𝑃{|𝑋₁|>𝜀𝑛^1/𝑞}𝑋≤𝐸(|¹𝜀|^𝑞𝑋ℎ(|¹𝜀|^𝑞))<∞. (2.35) With the help of the Markov inequality, we obtain 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘^′′−𝐸𝑆_𝑘^′′|>𝜀𝑛^1/𝑞2}≤4𝜀⁻²𝑛^−2/𝑞𝐸(max_{1≤𝑘≤𝑛}|𝑆_𝑘^′′−𝐸𝑆_𝑘^′′|²)≤8𝜀⁻²𝑛^−2/𝑞𝐸(𝑋^∗₁+⋯+𝑋^∗_𝑛)²=8𝜀⁻²𝑛^1−2/𝑞𝐸|𝑍₁−𝑍𝑌₁|². (2.36) From (2.34) and (2.35), it follows that (2.9) holds if the series ∑^∞_𝑛=1ℎ(𝑛)𝑛^−2/𝑞𝐸|𝑍₁−𝐸𝑍₁|² converges. This series can be estimated as follows: ^∞_𝑛=1ℎ(𝑛)𝑛^2/𝑞𝐸|𝑍₁−𝐸𝑍₁|²≤^∞_𝑛=1ℎ(𝑛)𝑛^2/𝑞𝐸|𝑍₁|²=^∞_𝑛=1ℎ(𝑛)𝑛^2/𝑞𝐸(𝑋²₁𝐼_{{|𝑋1|≤𝜀𝑛1/𝑞}})+𝜀^2∞_𝑘=1ℎ(𝑛)𝑃{|𝑋₁|>𝜀𝑛^1/𝑞}. (2.37) Rewrite the first summand on the right-hand side in the following way: ^∞_𝑛=1ℎ(𝑛)𝑛^2/𝑞𝐸(|𝑋₁|²𝐼_{{|𝑋1|≤𝜀𝑛1/𝑞}})=^∞_𝑛=1ℎ(𝑛)𝑛^𝑛2/𝑞_𝑘=1𝐸(𝑋²₁𝐼_{{𝜀(𝑘−1)}^1/𝑞_{<|𝑋1|≤𝜀𝑘}^1/𝑞_})=^∞∞_{𝑘=1𝑛=𝑘}ℎ(𝑛)𝑛^2/𝑞𝐸(𝑋²₁𝐼_{{𝜀(𝑘−1)}^1/𝑞_{<|𝑋1|≤𝜀𝑘}^1/𝑞_}). (2.38) From (2.1) and (2.2), it follows that there exist numbers 𝑘₀∈ℕ,𝑘₀>2, and 𝐶>0 such that 𝑘ℎ(𝑘)≤𝐶ℎ(2),^∞_𝑘ℎ(𝑥)𝑥^−2/𝑞𝑑𝑥≤𝐶ℎ(𝑘)𝑘^1−2/𝑞,∀𝑘≥𝑘₀. (2.39) For any 𝑘∈ℕ, we have ^∞_𝑛=𝑘ℎ(𝑛)𝑛^2/𝑞≤2^∞2/𝑞_{𝑛=𝑘𝑛}^𝑛+1ℎ(𝑥)𝑥^2/𝑞𝑑𝑥=2^2/𝑞∞_𝑘ℎ(𝑥)𝑥^2/𝑞𝑑𝑥<∞. (2.40) If 𝑘>𝑘₀, then ^∞_𝑛=𝑘ℎ(𝑛)𝑛^2/𝑞≤2^2/𝑞∞_𝑘ℎ(𝑥)𝑥^2/𝑞𝑑𝑥≤2^2/𝑞𝐶ℎ(𝑘)𝑘^1−2/𝑞≤2^2/𝑞𝐶²ℎ(𝑘−1)(𝑘−1)^1−2/𝑞. (2.41) With the help of these estimates, we get ^∞_𝑛=1ℎ(𝑛)𝑛^2/𝑞𝐸(|𝑋₁|²𝐼_{{|𝑋1|≤𝜀𝑛1/q}})≤2^𝑘2/𝑞0_𝑘=1^∞_𝑘ℎ(𝑥)𝑥^2/𝑞𝑑𝑥+2^2/𝑞𝐶^2∞_𝑘=𝑘0+1ℎ(𝑘−1)(𝑘−1)^2/𝑞−1𝐸(𝑋²₁𝐼_{{𝜀(𝑘−1)}^1/𝑞_{<|𝑋1|≤𝜀𝑘}^1/𝑞_}). (2.42) The last series can be estimated as follows: ^∞_𝑘=𝑘0+1ℎ(𝑘−1)(𝑘−1)^2/𝑞−1𝐸(𝑋²₁𝐼_{{𝜀(𝑘−1)}^1/𝑞_{<|𝑋1|≤𝜀𝑘}^1/𝑞_})≤𝜀²2^{∞2/𝑞−1}_𝑘=𝑘0+1𝐸(|𝜀⁻¹𝑋|^𝑞ℎ(|𝜀⁻¹𝑋₁|^𝑞)𝐼_{{𝜀(𝑘−1)}^1/𝑞_{<|𝑋1|≤𝜀𝑘}^1/𝑞_})≤𝜀²2^2/𝑞−1𝐸(|𝜀⁻¹𝑋₁|^𝑞ℎ(|𝜀⁻¹𝑋₁|^𝑞))<∞. (2.43) As a result we get that ∑^∞_𝑛=1ℎ(𝑛)𝑛^−2/𝑞𝐸(|𝑋₁|²𝐼_{{|𝑋1|≤𝜀𝑛1/𝑞}})<∞. Taking account of (2.35) and (2.37), we see that ∑^∞_𝑛=1ℎ(𝑛)𝑛^−2/𝑞𝐸|𝑍₁−𝐸𝑍₁|²<∞.
Prove that (2.9)⇒(2.8) for 𝑟=1. Note that^∞_𝑛=1ℎ(2^𝑛)𝑃{max_{2𝑛<𝑘≤2𝑛+1}|𝑋_𝑘|>2^𝑛/𝑞}≤^∞_𝑛=1ℎ(2^𝑛)𝑃{max_{2𝑛≤𝑘≤2𝑛+1}|𝑆_𝑘2|>^𝑛/𝑞2}≤2^∞2_𝑛=1^𝑛+2−1_𝑗=2𝑛+1ℎ(𝑗)𝑗𝑃{max_{2𝑛≤𝑘≤𝑗}|𝑆_𝑘2|>^−2/𝑞𝑗^1/𝑞2}≤2^∞_𝑛=1ℎ(𝑛)𝑛𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘2|>^−2/𝑞𝑛^1/𝑞2}<∞. (2.44) Denote 𝑏⁺_𝑛=𝑃{𝑋₁>2^𝑛/𝑞} and 𝑏⁻_𝑛=𝑃{−𝑋₁>2^𝑛/𝑞}. Note that 𝑃{|𝑋₁|>2^𝑛/𝑞}=𝑏⁺_𝑛+𝑏⁻_𝑛. With the help of (1.2), one can prove the inequality 1−(1−𝑏^±_𝑛)^2𝑛≤𝑃{max₂^𝑛+1_<𝑘≤2^𝑛+2|𝑋_𝑘|>2^𝑛/𝑞}. Since 𝑃{max₂^𝑛_<𝑘≤2^𝑛+1|𝑋_𝑘|>2^𝑛/𝑞}→0 and 𝑏^±_𝑛≤𝑃{|𝑋₁|>2^𝑛/𝑞}→0 as 𝑛→∞, then 2^𝑛ln(1−𝑏^±_𝑛)>−ln2 and 0≤𝑏⁺_𝑛+𝑏⁻_𝑛<1 for all 𝑛∈ℕ greater than some 𝑛₀. By the inequalities −ln(1−𝑥)≥𝑥 for 𝑥∈[0,1) and 1−𝑒^𝑥≥𝑒^𝑥|𝑥| for 𝑥≤0, we obtain 2^𝑛𝑏^±_𝑛≤−2^𝑛ln(1−𝑏^±_𝑛)≤2(1−𝑒^{2𝑛ln(1−𝑏±𝑛)})=2(1−(1−𝑏^±_𝑛)^2𝑛)≤2𝑃{max_{2𝑛+1<𝑘≤2𝑛+2}|𝑋_𝑘|>2^𝑛/𝑞},_∞𝑛=1ℎ(2^𝑛)(2^𝑛+1−2^𝑛)𝑃{|𝑋₁|>2^𝑛/𝑞}=^∞_𝑛=1ℎ(2^𝑛)2^𝑛(𝑏⁺_𝑛+𝑏⁻_𝑛)≤4^∞_𝑛=1ℎ(2^𝑛)𝑃{max_{2𝑛<𝑘≤2𝑛+1}|𝑋_𝑘|>2^𝑛/𝑞}<∞. (2.45) By Lemma 3.3, we have that 𝐸(|𝑋₁|^𝑞ℎ(|𝑋₁|^𝑞))<∞. In the same way as in the proof of the previous theorem, one can prove that 𝐸𝑋₁=0 if 𝑞≥1.

3. Auxiliary Results

Let 𝑋₁,…,𝑋_𝑛 be random variables. Denote 𝑆_𝑘=𝑋₁+⋯+𝑋_𝑘 for 𝑘=1,…,𝑛 and 𝐴_𝑛,𝛾=𝐸|𝑋₁|^𝛾+⋯+𝐸|𝑋_𝑛|^𝛾,𝐵_𝑛,𝛾=𝐸|𝑋₁−𝐸𝑋₁|^𝛾+⋯+𝐸|𝑋_𝑛−𝐸𝑋_𝑛|^𝛾 for some 𝛾∈(0,2],cosh𝑥=(𝑒^𝑥+𝑒^−𝑥)/2,𝑥∈ℝ=(−∞,∞).

Lemma 3.1. If negatively associated random variables 𝑋₁,…,𝑋_𝑛 are bounded by a constant 𝑐>0, then 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘𝑥|>𝑥}≤2exp{𝑐−𝑥𝑐ln(𝑥𝑐^𝛾−1𝐴_𝑛,𝛾+1)},0<𝛾≤1,𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘−𝐸𝑆_𝑘𝑥|>𝑥}≤4exp{2^𝛾−1𝑐−𝑥2𝑐ln(𝑥𝑐^𝛾−1𝐵_𝑛,𝛾+1)},1≤𝛾≤2, (3.1)(3.2) for any number 𝑥>0.

Proof. Prove Inequality (3.1). It is easily verified that {max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝑥}⊆{∑^𝑛_𝑘=1𝑋⁺_𝑘>𝑥}∪{∑^𝑛_𝑘=1𝑋⁻_𝑘>𝑥} where 𝑋⁺_𝑘=max{0,𝑋_𝑘} and 𝑋⁻_𝑘=max{0,−𝑋_𝑘}. With the help of Markov inequality, we get 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝑥}≤𝑃{^𝑛_𝑘=1𝑋⁺_𝑘∣>𝑥}+𝑃{^𝑛_𝑘=1𝑋⁻_𝑘>𝑥}≤𝑒^−ℎ𝑥𝐸exp{ℎ^𝑛_𝑘=1𝑋⁺_𝑘}+𝑒^−ℎ𝑥𝐸exp{ℎ^𝑛_𝑘=1𝑋⁻_𝑘} (3.3) for any ℎ>0. By Theorem A, random variables 𝑋⁺_𝑘,𝑘=1,…,𝑛, as well as 𝑋⁻_𝑘,𝑘=1,…,𝑛, are negatively associated. By Theorem B, the inequality 𝐸exp{ℎ(𝑋^±₁+⋯+𝑋^±_𝑛)}≤𝐸exp{ℎ((𝑋^∗₁)^±+⋯+(𝑋^∗_𝑛)^±)} (3.4) holds where random variables 𝑋^∗₁,…,𝑋^∗_𝑛 are independent, and for any 𝑘=1,…,𝑛, random variables 𝑋_𝑘 and 𝑋^∗_𝑘 are identically distributed. It follows that 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝑥}≤𝑒^{𝑛−ℎ𝑥}_𝑘=1𝐸𝑒^{ℎ(𝑋∗𝑘)+}+𝑒^{𝑛−ℎ𝑥}_𝑘=1𝐸𝑒^{ℎ(𝑋∗𝑘)−}=𝑒^{𝑛−ℎ𝑥}_𝑘=1𝐸𝑒^ℎ𝑋+𝑘+𝑒^{𝑛−ℎ𝑥}_𝑘=1𝐸𝑒^{ℎ𝑋−𝑘}≤2𝑒^{𝑛−ℎ𝑥}_𝑘=1𝐸𝑒^{ℎ|𝑋𝑘|}. (3.5) Further we can proceed as in Prokhorov [12], Fuk and Nagaev [13], and Kruglov [14]. Assume that 0<𝛾≤1. The function (𝑒^ℎ𝑥−1)/𝑥^𝛾,0≤𝑥≤𝑐, increases and hence (𝑒^ℎ𝑥−1)/𝑥^𝛾≤(𝑒^ℎ𝑐−1)/𝑐^𝛾. With the help of this inequality, we obtain ^𝑛_𝑘=1𝐸𝑒^{ℎ|𝑋𝑘|}=^𝑛_𝑘=1𝐸(1+|𝑋_𝑘|^−𝛾(𝑒^{ℎ|𝑋𝑘|}−1)|𝑋_𝑘|^𝛾)≤^𝑛_𝑘=1(1+𝑐^−𝛾(𝑒^ℎ𝑐−1)𝐸|𝑋_𝑘|^𝛾)=exp{𝑐^−𝛾(𝑒^ℎ𝑐−1)𝐴_𝑛,𝛾}. (3.6) From this inequality and from (3.5), it follows 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘|>𝑥}≤2exp{−ℎ𝑥+𝑐^−𝛾(𝑒^ℎ𝑐−1)𝐴_𝑛,𝛾}. (3.7) Put ℎ=𝑐⁻¹ln(𝑥𝑐^𝛾−1/𝐴_𝑛,𝛾+1). As a result we obtain (3.1).
Now we assume that 1≤𝛾≤2. By the Markov inequality we get𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘−𝐸𝑆_k|>𝑥}≤𝐸cosh(ℎmax_{1≤𝑘≤𝑛}|𝑆_𝑘−𝐸𝑆_𝑘|)cosh(ℎ𝑥) (3.8) for any ℎ>0. By Theorem B, the inequality 𝐸(max_{1≤𝑘≤𝑛}|𝑆_𝑘−𝐸𝑆_𝑘|)^𝑟≤2𝐸|(𝑋^∗₁−𝐸𝑋^∗₁)+⋯+(𝑋^∗_𝑛−𝐸𝑋^∗_𝑛)|^𝑟 (3.9) holds for any 𝑟>1. Denote 𝑆^∗_𝑛=𝑋^∗₁+⋯+𝑋^∗_𝑛. Note that cosh(|𝑥|)=cosh(𝑥) for any 𝑥∈ℝ. With these remarks, we obtain 𝐸cosh(ℎmax_{1≤𝑘≤𝑛}|𝑆_𝑘−𝐸𝑆_𝑘|)=1+^∞_𝑟=1𝐸(ℎmax_{1≤𝑘≤𝑛}|𝑆_k−𝐸𝑆_𝑘|^2𝑟)(2𝑟)!≤1+2^∞_𝑟=1𝐸|ℎ(𝑆^∗_𝑛−𝐸𝑆^∗_𝑛)|^2𝑟(2𝑟)!≤2𝐸cosh(ℎ|𝑆^∗_𝑛−𝐸𝑆^∗_𝑛|)=2𝐸cosh(ℎ(𝑆^∗_𝑛−𝐸𝑆^∗_𝑛)), (3.10) and hence 𝑃{max_{1≤𝑘≤𝑛}|𝑆_k−𝐸𝑆_𝑘|>𝑥}≤2𝐸cosh(ℎ(𝑆^∗_𝑛−𝐸𝑆^∗_𝑛))cosh(ℎ𝑥)≤2𝑒^−ℎ𝑥(𝐸𝑒^{ℎ(𝑆∗𝑛−𝐸𝑆∗𝑛)}+𝐸𝑒^{−ℎ(𝑆∗𝑛−𝐸𝑆∗𝑛)}). (3.11) By the inequalities 𝛼≤𝑒^𝛼−1 and 𝑒^𝛼−1−𝛼≤cosh𝛼−1 for 𝛼∈ℝ, we obtain 𝐸𝑒^{ℎ(𝑆∗𝑛−𝐸𝑆∗𝑛)}=^𝑛_𝑘=1𝐸𝑒^{ℎ(𝑋∗𝑘−𝐸𝑋∗𝑘)}≤^𝑛_𝑘=1exp{𝐸(𝑒^{ℎ(𝑋∗𝑘−𝐸𝑋∗𝑘)}−1)}=exp{^𝑛_𝑘=1𝐸(𝑒^{ℎ(𝑋∗𝑘−𝐸𝑋∗𝑘)}−1−ℎ(𝑋^∗_𝑘−𝐸𝑋^∗_𝑘))}≤exp{2^𝑛_𝑘=1𝐸(cosh(ℎ(𝑋^∗_𝑘−𝐸𝑋^∗_𝑘))−1)}. (3.12) Put 𝑓(𝛼)=(cosh𝛼−1)|𝛼|^−𝛾 for 𝛼≠0 and 𝑓(0)=1/2 if 𝛾=2, and 𝑓(0)=0 if 1≤𝛾<2. It can be easily verified that the function 𝑓 is continuous, even, and increases on (0,∞). Note that |𝑋^∗_𝑘−𝐸𝑋^∗_𝑘|≤2𝑐 for all 𝑘=1,…,𝑛. With these remarks, we obtain cosh(ℎ(𝑋^∗_𝑘−𝐸𝑋^∗_𝑘))−1=(cosh(ℎ(𝑋^∗_𝑘−𝐸𝑋^∗_𝑘))−1)|h(𝑋^∗_𝑘−𝐸𝑋^∗_𝑘)|^−𝛾|ℎ(𝑋^∗_𝑘−𝐸𝑋^∗_𝑘)|^𝛾≤(cosh(2ℎ𝑐)−1)(2𝑐)^−𝛾𝐸|𝑋^∗_𝑘−𝐸𝑋^∗_𝑘|^𝛾, (3.13) and hence 𝐸𝑒^{ℎ(𝑆∗𝑛−𝐸𝑆∗𝑛)}≤exp{(cosh(2ℎ𝑐)−1)(2𝑐)^−𝛾𝐵_𝑛,𝛾}. (3.14) In the same way, one can prove the inequality 𝐸𝑒^{−ℎ(𝑆∗𝑛−𝐸𝑆∗𝑛)}≤exp{(cosh(2ℎ𝑐)−1)(2𝑐)^−𝛾𝐵_𝑛,𝛾}. (3.15) From these inequalities and from (3.11), it follows that 𝑃{max_{1≤𝑘≤𝑛}|𝑆_𝑘−𝐸𝑆_𝑘|>𝑥}≤4exp{−ℎ𝑥+(cosh(2ℎ𝑐)−1)(2𝑐)^−𝛾𝐵_𝑛,𝛾}≤4exp{−ℎ𝑥+2(𝑒^2ℎ𝑐−1)(2𝑐)^−𝛾𝐵_𝑛,𝛾}. (3.16) Put ℎ=(2𝑐)⁻¹ln(𝑥𝑐^𝛾−1/𝐵_𝑛,𝛾+1). As a result we obtain (3.2).

Lemma 3.2. Let ℎ(𝑥),𝑥∈[0,∞), be a nondecreasing nonnegative function, 𝜉 be a nonnegative random variable, 𝑟≥1,𝑎𝑛𝑑𝑞>0. Then ^∞_𝑛=1𝑛^𝑟−1ℎ(𝑛)𝑃{𝜉>𝑛^1/𝑞}≤𝐸(𝜉^𝑟𝑞ℎ(𝜉^𝑞)). (3.17)

Proof. It can be found in Kruglov and Volodin [7].

Lemma 3.3. Let ℎ(𝑥),𝑥∈[0,∞), be a nondecreasing nonnegative function possessing property (2.1), 𝜉 be a nonnegative random variable, 𝑟≥1,𝑞>0, and {𝑏_𝑛}_𝑛≥1 an unbounded nondecreasing sequence of positive numbers such that 𝑏_𝑛+1≤𝑏𝑏_𝑛 for all 𝑛∈ℕ and for some number 𝑏>0,𝑏₀=0. Then there exist numbers 𝑘₀∈ℕ and 𝑑>0 such that 𝑑𝐸(𝜉^𝑟𝑞ℎ(𝜉^𝑞))𝐼_{{𝜉≥𝑏𝑘0}})≤^∞_𝑛=1𝑏_𝑛^𝑟−1ℎ(𝑏_𝑛)(𝑏_𝑛−𝑏_𝑛−1)𝑃{𝜉>𝑏_𝑛^1/𝑞}. (3.18)

Proof. It can be found in Kruglov and Volodin [7].

The next lemma was proved in Kruglov and Volodin [7] under an additional restriction.

Lemma 3.4. Let 𝑋_𝑛,𝑛∈ℕ, be identically distributed random variables such that 𝐸|𝑋₁|^𝑞<∞ for some 𝑞∈(0,2) and 𝐸𝑋₁=0 if 1≤𝑞<2. Define the function 𝑓(𝑥)=−𝜀𝑛^1/𝑞𝐼_{(−∞,−𝜀𝑛1/𝑞)}(𝑥)+𝑥𝐼_{[−𝜀𝑛1/𝑞,𝜀𝑛1/𝑞]}(𝑥)+𝜀𝑛^1/𝑞𝐼_{(𝜀𝑛1/𝑞,∞)}(𝑥),𝑥∈ℝ, where 𝜀>0 is a fixed number. Then lim_𝑛→∞1𝑛^1/𝑞max_{1≤𝑘≤𝑛}|^𝑘_𝑗=1𝐸𝑓(𝑋_𝑗)|=0. (3.19)

Proof. Note that max_{1≤𝑘≤𝑛}|^𝑘_𝑗=1𝐸𝑓(𝑋_𝑗)|=𝑛|𝐸𝑓(𝑋₁)|≤n|𝐸(𝑋₁𝐼_{{|𝑋1|≤𝜀𝑛1/𝑞}})|+𝜀𝑛^1/𝑞𝑃{|𝑋₁|>𝜀𝑛^1/𝑞},lim_𝑛→∞1𝑛^1/𝑞𝑛^1/𝑞+1𝑃{|𝑋₁|>𝜀𝑛^1/𝑞}=lim_𝑛→∞𝑛𝑃{|𝑋₁|^𝑞>𝜀^𝑞𝑛}=0. (3.20) It suffices to prove that lim_𝑛→∞𝑛𝑛^1/𝑞𝐸(𝑋₁𝐼_{{|𝑋1|≤𝜀𝑛1/𝑞}})=0. (3.21) Suppose that 0<𝑞<1. For any 𝛿>0, there exist 𝑛₀∈ℕ such that 𝐸(|𝑋₁|^𝑞𝐼_{{|𝑋1|>𝜀𝑛1/𝑞}})<𝛿. We get, for any 𝑛≥𝑛₀,𝑛𝑛^1/𝑞𝐸(|𝑋₁|𝐼_{{|𝑋1|≤𝜀𝑛1/𝑞}}𝑛)≤𝑛^1/𝑞𝐸(|𝑋₁|𝐼_{{|𝑋1|≤𝜀𝑛0}^1/𝑞_})+𝜀^1−𝑞𝐸(|𝑋₁|^𝑞𝐼_{𝜀𝑛0^1/𝑞_{<|𝑋1|≤𝜀𝑛1/𝑞}}), (3.22) and hence limsup_𝑛→∞𝑛𝑛^1/q𝐸(|𝑋₁|𝐼_{{|𝑋1|≤𝑛1/𝑞}})≤𝜀^1−𝑞𝛿. (3.23) This implies (3.21), since 𝛿>0 can be chosen arbitrarily small.
If 1≤𝑞<2 and 𝐸𝑋₁=0, then we get𝑛𝑛^1/𝑞|𝐸(𝑋₁𝐼_{{|𝑋1|≤𝜀𝑛1/𝑞}}𝑛)|=𝑛^1/𝑞|𝐸(𝑋₁𝐼_{{|𝑋1|>𝜀𝑛1/𝑞}})|≤𝜀^1−𝑞𝐸(|𝑋₁|^𝑞𝐼_{{|𝑋1|>𝜀𝑛1/𝑞}})→0. (3.24)

Acknowledgments

The author is grateful to an anonymous referee for careful reading of the paper and useful remarks. This research was supported by the Russian Foundation for Basic Research, projects 08-01-00563-a, 08-01-00567-a, 08-01-90252 Uzbekistan.