The almost sure local central limit theorem for products of partial sums under negative association

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{X_{n}, n\geq1\}$\end{document}{Xn,n≥1} be a strictly stationary negatively associated sequence of positive random variables with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{E}X_{1}=\mu>0$\end{document}EX1=μ>0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{Var}(X_{1})=\sigma^{2}<\infty$\end{document}Var(X1)=σ2<∞. Denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_{n}=\sum_{i=1}^{n}X_{i}, p_{k}=\mathrm{P}(a_{k}\leq ({\prod}_{j=1}^{k}S_{j}/(k!\mu^{k}) )^{1/(\gamma\sigma_{1} \sqrt{k})}< b_{k})$\end{document}Sn=∑i=1nXi,pk=P(ak≤(∏j=1kSj/(k!μk))1/(γσ1k)<bk) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma=\sigma/\mu$\end{document}γ=σ/μ the coefficient of variation. Under some suitable conditions, we derive the almost sure local central limit theorem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim_{n\rightarrow\infty}\frac{1}{\log n}\sum_{k=1}^{n} \frac{1}{kp_{k}}\mathrm{I} \biggl\{ a_{k}\leq \biggl(\frac {\prod_{j=1}^{k}S_{j}}{k!\mu^{k}} \biggr)^{1/(\gamma\sigma_{1} \sqrt {k})}< b_{k} \biggr\} =1 \quad\mbox{a.s.,} $$\end{document}limn→∞1logn∑k=1n1kpkI{ak≤(∏j=1kSjk!μk)1/(γσ1k)<bk}=1a.s., where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma_{1}^{2}=1+\frac{1}{\sigma^{2}}\sum_{j=2}^{\infty}\operatorname{Cov}(X_{1},X_{j})>0$\end{document}σ12=1+1σ2∑j=2∞Cov(X1,Xj)>0.

where σ 2 1 = 1 + 1 σ 2 1 Introduction Definition 1.1 ([1]) A finite family of random variables X 1 , X 2 , . . . , X n , n ≥ 2, is said to be negatively associated (NA) if, for every pair of disjoint subsets A and B of {1, 2, . . . , n}, we have where f 1 and f 2 are coordinatewise increasing and the covariance exists. An infinite family of random variables (r.v.) is NA if every finite subfamily is NA.
Obviously, if {X i , i ≥ 1} is NA, and {f i , i ≥ 1} is a sequence of nondecreasing (or nonincreasing) functions, then {f i (X i ), i ≥ 1} is also NA. We refer to Roussas [2] for NA's fundamental properties and applications in several fields, Shao [3] for the moment inequalities, Jing and Liang [4] and Cai [5] for the strong limit theorems, Chen et al. [6] and Sung [7] for the complete convergence.
Let S n := n i=1 X i denote the partial sum of {X i , i ≥ 1} and n j=1 S j is known as a product of partial sum S j , the study on partial sum has received extensive attention. Such well-known classic laws as the central limit theorem (CLT), the almost sure central limit theorem (AS-CLT), and law of the iterated logarithm (LIL) are known for characterizing the asymptotic behavior of S n . However, the study of asymptotic behavior for product of partial sum is not so far, it was initiated by Arnold and Villaseñor [8]. This paper intends to study the limit behavior of product n j=1 S j under negative association. Let {X n , n ≥ 1} be a strictly stationary NA sequence of positive r.v. with EX 1 = μ > 0, Var(X 1 ) = σ 2 < ∞, and the coefficient of variance γ = σ /μ. Assume that 1. Li and Wang [9] obtained the following version of the CLT: where N is a standard normal distribution random variable. 2. Li and Wang [10] proved the following ASCLT: here and elsewhere, I{A} represents the indicative function of the event A and F(·) is the distribution function of the log-normal random variable exp( √ 2N ). The almost sure central limit theorem was proposed by Brosamler [11] and Schatte [12]. In recent years, the ASCLT has been extensively studied, and an attractive research direction is to prove it under associated or dependent situations. There are some literature works for α, ρ, φ-mixing and associated random variables, we refer to Matuła [13], Lin [14], Zhang et al. [15], Matuła and Stȩpień [16], Hwang [17], Li [18], Miao and Xu [19], Wu and Jiang [20].
A more general version of ASCLT for products of partial sums was proved by Weng et al. [21]. The following theorem is due to them.
Theorem A Let {X n , n ≥ 1} be a sequence of independent and identically distributed positive random variables with EX 3 and assume for sufficiently large k, p k ≥ 1/(log k) δ 1 for some δ 1 > 0. Then we have This result may be called almost sure local central limit theorem (ASLCLT) for the product n j=1 S j of independent and identically distributed positive r.v., while (1.4) may be called almost sure global central limit theorem (ASGCLT).
The ASLCLT for partial sums of independent and identically distributed r.v. was stimulated by Csáki et al. [22], and Khurelbaatar [23] extended it to the case of ρ-mixing sequences, Jiang and Wu [24] extended it to the case of NA sequences. Zang [25] obtained the ASLCLT for a sample range.
In this paper, our concern is to give a common generalization of (1.7) to the case of NA sequences. The remainder of the paper is organized as follows. Section 2 provides our main result. Section 3 gives some auxiliary lemmas. The proofs of the theorem and some lemmas are in Sect. 4.

Main results
In the following, we assume that {X n , n ≥ 1} is a strictly stationary negatively associated sequence of positive r.v. 's with EX 1 = μ > 0, Var(X 1 ) = σ 2 < ∞, EX 3 1 < ∞, the coefficient of and Then we study the asymptotic behavior of the logarithmic average where the expression in the sum above is defined to be one if the denominator is zero. That is, let {a n , n ≥ 1} and {b n , n ≥ 1} be two sequences of real numbers and Therefore, we should study the asymptotic limit properties of 1 log n n k=1 α k k under suitable conditions.
In the following discussion, we shall use the definition of the Cox-Grimmett coefficient and we can verify that the formula is correct for a stationary sequence of negatively associated random variables. In the following, ξ n ∼ η n denotes ξ n /η n → 1, n → ∞. ξ n = O(η n ) denotes that there exists a constant c > 0 such that ξ n ≤ cη n for sufficiently large n. The symbols c, c 1 , c 2 , . . . represent generic positive constants.
Obviously (2.9) holds, then (2.10) becomes (1.4), which is the ASGCLT. Thus the ASLCLT is a general result which contains the ASGCLT.

Auxiliary lemmas
In order to prove the main theorem, we need to use the concept of a triangular array of random variables. Let b k,n = n i=k 1/i and Y i = (X iμ)/σ . We define a triangular array Z 1,n , Z 2,n , . . . , Z n,n as Z k,n = b k,n Y k and put S k,n = Z 1,n + Z 2,n + · · · + Z k,n for 1 ≤ k ≤ n. Let where Note that, for l > k, we have So, by the property of NA sequences, S l,l -S k,kb k+1,l S k and U k are negatively associated. The following Lemma 3.1 is due to Liang et al. [26].
be a sequence of NA random variables with EX n = 0 and {a ni , 1 ≤ i ≤ n, n ≥ 1} be an array of real numbers such that sup n n i=1 a 2 ni < ∞ and max 1≤i≤n |a ni | → 0 as n → ∞. Assume that j:|k-j|≥n | Cov(X k , X j )| → 0 as n → ∞ uniformly for k ≥ 1. If Var( n i=1 a ni X i ) = 1 and {X 2 n , n ≥ 1} is a uniformly integrable family, Now we obtain the CLT for triangular arrays.
Suppose that there exist constants δ 2 and δ 3 such that 0 < δ 2 , δ 3 < 1. Assume also that (1.1) and (1.2) hold. If for sufficiently large n, then The proof is quite long and will be left to Sect. 4. The following Lemma 3.3 is a corollary to Corollary 2.2 in Matuła [27] under a strictly stationary condition.
Then we have
The proof will be left to Sect. 4. The following result is due to Khurelbaatar [23]. The following Lemma 3.6 is obvious.

Proofs of the main result and lemmas
The main aspect of our proof of Theorem 2.1 is verification condition (3.13) for α k , where α k is defined by (2.5). We use ASCLT (1.4) with remainders and the following elementary inequalities: with some constant c. Moreover, for each k > 0, there exists c 1 = c 1 (k) such that Proof of Theorem 2.1 Let Thus, -∞ ≤â k ≤ 0 ≤b k ≤ ∞ by (2.1). By the definition of U k in (3.1), we have p k = P(â k ≤ U k <b k ) and Var(α k ) where δ 2 , δ 3 are defined by Lemma 3.2. Note also that Var(α k ) = 0 if p k = 0 and And by the condition of (2.9), we have If either p k = 0 or p l = 0, then obviously Cov(α k , α l ) = 0, so we may assume that p k p l = 0, by (2.1), we have for δ 1 < 1/4 and δ 2 < 7/8. Now we estimate the bound of 3 . Let A n be an integer such that log A n ∼ (log n) δ 2 for sufficiently large n. Then So, it remains to estimate the bound of 4 . Let 1 ≤ k < l and ε l = 1/(log l) δ 4 , where 0 < δ 1 < So by (3.3), Lemma 3.3, and (4.1), we obtain So, by using Lemma 3.4, we have Hence applying Lemma 3.5, our theorem is proved under the restricting condition (4.5). Then, we remove the restricting condition (4.5). Fix x > 0 and define Clearly b ka k ≤ min(2x, c) and p k ≤ p k , so assuming p k = 0, then we also have p k = 0, thus . (4.11) By the law of large numbers, we get ( S i iμ -1) P → 0. Noting that x 2 /(1 + θ x) 2 ≤ 4x 2 for |x| < 1/2 and θ ∈ (0, 1), and by using Markov's inequality, ∀ε > 0, we have Thus, we obtain and a.s.