Precise Rates in Log Laws for NA Sequences

Let X1,X2,… be a strictly stationary sequence of negatively associated (NA) random variables with EX1=0, set Sn=X1


Introduction
A finite family of random variables, X 1 ,X 2 ,...,X n , is said to be NA if, for every pair of disjoint subsets T 1 and T 2 of {1, 2,...,n}, whenever f 1 and f 2 are coordinatewise increasing and the covariance exists.An infinite family is NA if every finite subfamily is NA.This definition was introduced by Alam and Saxena [1] and Joag-Dev and Proschan [2], and has found many applications in percolation theory, multivariate statistical analysis, and reliability theory (see, e.g., Barlow and Proschan [3]).
Let {X n : n ≥ 1} be a sequence of NA random variables on some probability space (Ω,Ᏺ,P) with mean zero and finite variance.As usual, set S 0 = 0, S n = X 1 + ••• + X n , n ≥ 1, and write σ 2 n = ES 2 n .Under appropriate covariance conditions, many limit theorems have been obtained.For example, the central limit theorem was proved by Newman [4].
Note that in the above-mentioned limit theorems, the convergence rates of logarithm are little known, the purpose of the present paper is to investigate the precise asymptotics in the law of the logarithm for NA sequences.It is well known that NA sequences can contain independent random variables as special case, many authors have given lots of beautiful results for independent variables.Let us first recall parts of those results, it is very convenient to adopt the following notations: let X 1 ,X 2 ,... be independent and identically distributed (i.i.d.) nondegenerate random variables with EX 1 = 0 and Chow and Lai [13] studied the following results.Theorem 1.2.Suppose that Var X 1 = σ 2 and α ≥ 1.Then the following are equivalent: (1.4) Heyde [14] presented an interesting and beautiful result.
This is a precise estimate for the convergence rate of probability series as ↓ 0, which has been generalized and extended in several directions.For α = 1 in Theorem 1.2, Gut and Sp ǎtaru [15] obtained the results as follows.
Yuexu Zhao 3 Theorem 1.4.Suppose that EX 1 = 0 and EX where N is a standard normal random variable.
Our starting point is Theorem 1.4, the present work will give the analogue of (1.6) for NA sequences.From now on, we adopt the following notations: let X 1 ,X 2 ,... be strictly stationary NA sequences with write log for the natural logarithm, log x = log e (x ∨ e), [z] denotes the largest integer which is not larger than z, C denotes positive constant, independent of , it may take different values in each appearance.The paper is organized as follows: we first introduce our main results, after which the proofs of Theorems 2.1 and 2.4 are exposed in Sections 3 and 4, respectively.We now state the main results.

Main results
where N is a standard normal random variable.
Corollary 2.2.Under the conditions in Theorem 2.1, (2.4) Without loss of generality, throughout the paper, we will suppose that σ 2 = 1.Let Φ(x) denote the standard normal distribution function, and put

Proof of Theorem 2.1
In order to prove this result easily, we separate the proof into two propositions, the first one can be formulated as follows.
Proposition 3.1.Suppose that N be a nondegenerate Gaussian random variable.Then Proof.Noting the definition of κ n , we first show that lim By integral formula and transformation, it is enough to show that for any α > −1, The proof of (3.1) should be completed, if one could show that lim the proof of (3.5) is similar to that of Proposition 2.2 in Huang and Zhang [16].Before giving the second proposition, the following lemma is necessary.
Proof of Theorem 2.1.Combining Propositions 3.1 and 3.3, one can complete the proof of this theorem immediately.

Proof of Theorem 2.4
The following propositions will simplify the proof of Theorem 2.4, which are stated as follows.
Proposition 4.1.Suppose that {W (t) : t ≥ 0} be a standard Wiener process (Brownian motion).Then Proof.Noting the result of Billingsley [18], where N is the standard normal random variable.Then, according to Proposition 3.1, one can complete the proof easily.Lemma 4.2 [7].Suppose that {X n : n ≥ 1} be strictly stationary NA sequences, EX 1 = μ, 0 < Var X 1 = σ 2 < ∞, and where W(t) is the standard Wiener process and C[0,T] is the usual C space on [0,T].