1 Introduction
Statistical test depends greatly on sampling, and the random sampling without replacement from a finite population is negatively associated (NA), but it is not independent. The concept of NA was introduced by Joag-Dev and Proschan [
1] in which the fundamental properties were studied. Limit behaviors of NA have received increasing attention recently due to the wide applications of NA sampling in a lot of fields such as those in multivariate statistical analysis and reliability. Scholars have also achieved some results. For example, Shao [
2] for the moment inequalities, Su and Wang [
3] for Marcinkiewicz-type strong law of large number. Specifically, the definition of NA random variables is as follows.
Starting with Arnold and Villaseñor [
4], asymptotic properties of the products of partial sums were investigated by several authors in the last two decades. Arnold and Villaseñor [
4] discussed sums of records and gave the result that the products of i.i.d. positive, square integrable random variables are asymptotically log-normal. After that, Rempała and Wesołowski [
5] removed the condition with exponential distribution in Arnold and Villaseñor [
4] and introduced central limit theorem (CLT) for the product of partial sums. Miao [
6], in different perspective, gave a new form of the product of partial sums. Later, Xu and Wu [
7] generalized the result of Miao [
6] from the case of i.i.d. random variables to NA random variables and obtained the following result. Let
\(\{X_{n},n\geq1\}\) be a strictly stationary negatively associated sequence of positive random variables with
\(\mathbb{E} X_{1}=\mu\),
\(\sigma^{2}={\mathbb{E}(X_{1}-\mu)^{2}}+ 2\sum_{k=2}^{\infty}\mathbb{E}(X_{1}-\mu) (X_{k}-\mu)>0\). Then
$$\begin{aligned} \biggl(\frac{\prod_{i=1}^{k}S_{k,i}}{(k-1)^{k}\mu^{k}} \biggr)^{\mu/(\sigma \sqrt{k})} \overset{d}{ \longrightarrow} e^{\mathcal{N}} \quad\text{as} k\rightarrow \infty, \end{aligned}$$
(1)
where
\(S_{k,i}=\sum_{j=1}^{k}X_{j}-X_{i}\), and
\(\mathcal{N}\) is a standard normal random variable.
During the past two decades, several researchers also focused on the almost sure central limit theorem (ASCLT) for the partial sums
\(S_{k}/\sigma_{k}\) of random variables which was started by Brosamler [
8] and Schatte [
9]. For the product of partial sums, Gonchigdanzan and Rempała [
10] and Miao [
6] obtained some results related to (
1). Xu and Wu [
7] also generalized the result of Miao [
6] not only from the case of i.i.d. random variables to NA random variables but also from weight
\(d_{k}=1/k\) to
\(d_{k}=\log(c_{k+1}/c_{k})\exp(\log^{\alpha}k)\),
\(0\le\alpha<1/2\), where
\(0\le c_{k}\to\infty,\lim_{k\to\infty}c_{k+1}/c_{k}=c<\infty\). That is, for any real
x,
$$\begin{aligned} \lim_{n\rightarrow\infty}\frac{1}{D_{n}} \sum _{k=1}^{n} {d_{k}} \text{I} \biggl( \biggl( \frac{\prod_{i=1}^{k}S_{k,i}}{(k-1)^{k}\mu^{k}} \biggr)^{\mu /(\sigma\sqrt{k})} \le x \biggr) = F(x) \quad\text{a.s.}, \end{aligned}$$
(2)
where
\(D_{n}=\sum_{k=1}^{n}d_{k}\),
\(\text{I}(\cdot)\) denotes the indicator function and
\(F(x)\) is the distribution function of the random variable
\(e^{\mathcal{N}}\). Since then, scholars have generalized these results for weight sequence or/and the range of random variables. For instance, Wu [
11] extended weight sequence from
\(1/k\) to
\(d_{k}=k^{-1}e^{\ln^{\alpha}k}\),
\(0\le\alpha<1\). Tan et al. [
12] extended the range of random variables from i.i.d. random variables to
\(\rho^{-}\)-mixing sequences and Ye and Wu [
13] extended i.i.d. random variables to strongly mixing random variables. We refer the reader to Berkes and Csáki [
14], Hörmann [
15], Wu [
16], Xu and Wu [
17], and Tan and Liu [
18] for ASCLT.
A more general version of ASCLT was proved by Csáki et al. [
19] who proved
$$\begin{aligned} \lim_{n\rightarrow\infty}\frac{1}{\log n}\sum _{k=1}^{n} \frac{\text{I}(a_{k}\le S_{k}< b_{k})}{k\mathbb{P}(a_{k}\le S_{k}< b_{k})}=1 \quad\text{a.s.}, \end{aligned}$$
(3)
under the conditions
\(-\infty\le a_{k}\le0\le b_{k}\le\infty\),
\(\mathbb {E}|X_{1}|^{3}<\infty\), and
$$\begin{aligned} \sum_{k=1}^{n}\frac{\log k}{k^{3/2}\mathbb{P}(a_{k}\le S_{k}< b_{k})}=O(\log n) \quad\text{as } n\rightarrow\infty. \end{aligned}$$
The result above may be called almost sure local central limit theorem. Gonchigdanzan [
20] and Jiang and Wu [
21] extended (
3) to the case of
ρ-mixing sequences and NA sequences, respectively. Weng et al. [
22] proved the almost sure local central limit theorem for the product of partial sums of independent and identically distributed positive random variables. Recently, Jiang and Wu [
23] extended the result in Weng et al. [
22] from i.i.d. to NA sequences.
In this paper, our objective is to give the almost sure local central limit theorem for the product of some partial sums of NA sequences related to (
2).
This paper is organized as follows. The exact result is described in Sect.
2. In Sect.
3 some auxiliary lemmas are provided. Proofs are presented in Sect.
4.
2 Main result
In the following, let
c be a positive constant to vary from one place to another.
\(a_{n}\sim b_{n}\) denotes
\(\lim_{n\to\infty}a_{n}/b_{n}=1\). Assume that
\(\{X_{n},n\geq1\}\) is a strictly stationary sequence of NA random variables with
\({\mathbb{E}} X_{1}=\mu\),
\(0<\operatorname{Var}X_{1}<\infty\). Denote
$$\begin{aligned} S_{k,i}=\sum_{j=1}^{k}X_{j}-X_{i} \quad\text{for } 1\le i\le k,\quad\quad Y_{j} = {X_{j}-\mu} \quad\text{for } j\geq1, \qquad\widetilde{S}_{k} = \sum _{j=1}^{k} Y_{j}, \end{aligned}$$
and the covariance structure of the sequences
$$\begin{aligned} u(k)= \sup_{j\in\mathbb{N}}\sum_{i: \vert i-j \vert \geq k} \bigl\vert \operatorname {Cov}(X_{i},X_{j}) \bigr\vert ,\quad k\in \mathbb{N}\cup\{0\}. \end{aligned}$$
For a stationary sequence of NA random variables, we point out
$$\begin{aligned} u(k)=-2\sum_{j=k+1}^{\infty}\operatorname{Cov}(X_{1},X_{j}), \quad k\in N. \end{aligned}$$
By Lemma 8 of Newman [
24], we know
\(u(0)<\infty\) and
\({\lim}_{k\to\infty}u(k)=0\). From Newman [
25],
\(\sigma^{2}:=\mathbb{E}Y_{1}^{2}+ 2\sum_{k=2}^{\infty}\mathbb{E}Y_{1}Y_{k}\) always exists and
\(\sigma^{2}\in[0,\operatorname{Var}Y_{1}]\). Further, if
\(\sigma^{2}>0\), then
\(\operatorname{Var}\widetilde {S}_{k}:=\sigma_{k}^{2}\sim k\sigma^{2}\).
Let
\(\{a_{k},k\geq1\}\) and
\(\{b_{k},k\geq1\}\) be two sequences of real numbers satisfying
$$\begin{aligned} 0\le a_{k}\le1 \le b_{k} \le\infty,\quad k=1,2, \ldots. \end{aligned}$$
(4)
Set
$$\begin{aligned} p_{k}=\mathbb{P} \biggl(a_{k}\le \biggl(\frac{\prod_{i=1}^{k}S_{k,i}}{(k-1)^{k}\mu^{k}} \biggr)^{\mu/(\sigma\sqrt{k})} < b_{k} \biggr) \end{aligned}$$
and
$$\begin{aligned} \alpha_{k}= \textstyle\begin{cases} \frac{1}{p_{k}}\text{I} (a_{k}\le (\frac{\prod_{i=1}^{k}S_{k,i}}{(k-1)^{k}\mu^{k}} )^{\mu/(\sigma\sqrt{k})}< b_{k} ), &\text{if } p_{k}\neq0,\\ 1, &\text{if } p_{k}= 0. \end{cases}\displaystyle \end{aligned}$$
(5)
Our main result is as follows.
3 Lemmas
Let
\(C_{k,i} = {S_{k,i}}/{((k-1)\mu)}\),
\(k=1,2,\dots\). By the following logarithm Taylor expansion
$$\begin{aligned} \log(x+1)=x-\frac{x^{2}}{2(1+\theta x)^{2}}, \end{aligned}$$
where
\(\theta\in(0,1)\) depends on
\(x\in(-1,1)\). Denote
$$\begin{aligned} U_{k} =& \frac{\mu}{\sigma\sqrt{k}}\sum_{i=1}^{k} \log\frac{S_{k,i}}{(k-1)\mu} = \frac{\mu}{\sigma\sqrt{k}}\sum_{i=1}^{k} \log C_{k,i} \\ =& \frac{\mu}{\sigma\sqrt{k}}\sum_{i=1}^{k} \biggl( (C_{k,i}-1 )-\frac{(C_{k,i}-1)^{2}}{2 (1+\theta_{i}(C_{k,i}-1) )^{2}} \biggr) \\ :=&\frac{1}{\sigma\sqrt{k}}\widetilde{S}_{k} + T_{k}, \end{aligned}$$
where
$$\begin{aligned} T_{k} = \frac{\mu}{2\sigma\sqrt{k}}\sum _{i=1}^{k}\frac{(C_{k,i}-1)^{2}}{(1+\theta_{i}(C_{k,i}-1) )^{2}},\quad \theta_{i} \in(0,1). \end{aligned}$$
In order to prove the main result, the following lemmas play important roles in the proof of our theorem. The following result is due to Weng et al. [
22].
The following Lemma
2 is Marcinkiewicz-type strong law of large numbers given by Su and Wang [
3] for identically distributed NA sequences.
Lemma
3 is from Corollary 2.2 in Matuła [
26] due to NA random variables which are linearly negative quadrant dependent (LNQD). Of course it is the Berry–Esseen inequality for the NA sequence random variables, which is also studied in Pan [
27].
The following Lemma
4 is obvious.
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