An almost sure central limit theorem of products of partial sums for ρ^--mixing sequences

For a random variable X, define ∥X∥ _p = (E|X| ^p )^1/p. For two nonempty disjoint sets S,T ⊂ N, we define dist(S,T) to be min{|j - k|; j ∈ S, k ∈ T}. Let σ(S) be the σ-field generated by {X _k , k ∈ S}, and define σ(T) similarly. Let C be a class of functions which are coordinatewise increasing. For any real number x, x⁺, and x^- denote its positive and negative part, respectively, (except for some special definitions, for examples, ρ^-(s), ρ^-(S,T), etc.). For random variables X, Y, define

where the sup is taken over all f , g ∈ C such that E(f(X))² < ∞ and E(g(Y))² < ∞.

A sequence {X _n , n ≥ 1} is called negatively associated (NA) if for every pair of disjoint subsets S, T of N,

whenever f , g ∈ C.

A sequence {X _n , n ≥ 1} is called ρ*-mixing if

where

A sequence {X _n , n ≥ 1} is called ρ^--mixing, if

where,

The concept of ρ^--mixing random variables was proposed in 1999 (see [1]). Obviously, ρ^--mixing random variables include NA and ρ*-mixing random variables, which have a lot of applications, their limit properties have aroused wide interest recently, and a lot of results have been obtained, such as the weak convergence theorems, the central limit theorems of random fields, Rosenthal-type moment inequality, see [1‐4]. Zhou [5] studied the almost sure central limit theorem of ρ^--mixing sequences by the conditions provided by Shao: on the conditions of central limit theorem, and if ε 0 > 0 , V a r ∑ i = 1 n 1 i f S i σ i = O log 2 - ε 0 n, where f is Lipschitz function. In this article, we study the almost sure central limit theorem of products of partial sums for ρ^--mixing sequence by the central limit theorem of weighted sums and moment inequality.

Here and in the sequel, let b k , n = ∑ i = k n 1 i , k ≤ n with b_k,n= 0, k > n. Suppose {X _n , n ≥ 1} be a strictly stationary ρ^--mixing sequence of positive random variables with EX₁ = μ > 0 and Var(X₁) = σ² < ∞. Let S ̃ n = ∑ k = 1 n Y k and S n , n = ∑ k = 1 n b k , n Y k, where Y k = X k - μ σ , k ≥ 1. Let σ n 2 = V a r S n , n, and C denotes a positive constant, which may take different values whenever it appears in different expressions. The following are our main results.

Theorem 1.1 Let {X _n , n ≥ 1} be a defined as above with 0 < E|X₁| ^r < ∞ for a certain r > 2, denote S n = ∑ i = 1 n X i and γ = σ μ the coefficient of variation. Assume that

(a₁) σ 1 2 = E X 1 2 + 2 ∑ n = 2 ∞ C o v X 1 , X n > 0,

(a₂) ∑ n = 2 ∞ C o v X 1 , X n < ∞,

(a₃) ρ^-(n) = O(log^-δn), ∃ δ > 1,

(a₄) inf n ∈ N σ n 2 n > 0.

Then

Here and in the sequel, I{·} denotes indicator function and Φ(·) is the distribution function of standard normal random variable N.

Theorem 1.2 Under the conditions of Theorem 1.1, then

Here and in the sequel, F(·) is the distribution function of the random variable e 2 N.

To prove our main results, we need the following lemmas.

Lemma 2.1 [3] Let {X _n , n ≥ 1} be a weakly stationary ρ^--mixing sequence with E X n = 0 , 0 < E X 1 2 < ∞, and

then

Lemma 2.2 [4] For a positive real number q ≥ 2, if {X _n , n ≥ 1} is a sequence of ρ^--mixing random variables with EX _i = 0, E|X _i | ^q < ∞ for every i ≥ 1, then for all n ≥ 1, there is a positive constant C = C(q, ρ^-(·)) such that

Lemma 2.3 [6] ∑ i = 1 n b i , n 2 = 2 n - b 1 , n.

Lemma 2.4 [[3], Theorem 3.2] Let {X _ni , 1 ≤ i ≤ n, n ≥ 1} be an array of centered random variables with E X n i 2 < ∞ for each i = 1,2,...,n. Assume that they are ρ^--mixing. Let {a _ni , 1 ≤ i ≤ n, n ≥ 1} be an array of real numbers with a _ni = ±1 for i = 1, 2,..., n. Denote σ n 2 = V a r ∑ i = 1 n a n i X n i and suppose that

and

and the following Lindeberg condition is satisfied:

for every ε > 0. Then

Lemma 2.5 Let {X _n , n ≥ 1} be a strictly stationary sequence of ρ^--mixing random variables with EX _n = 0 and ∑ n = 2 ∞ C o v X 1 , X n < ∞ , a n i , 1 ≤ i ≤ n , n ≥ 1 be an array of real numbers such that sup n ∑ i = 1 n a n i 2 < ∞ and max 1 ≤ i ≤ n a n i → 0 as n → ∞. If V a r ∑ i = 1 n a n i X i = 1 and X n 2 is an uniformly integrable family, then

Proof Notice that

where b n i = a n i a n i and Y _ni = |a _ni |X _i . Then {Y _ni , 1 ≤ i ≤ n, n ≥ 1} is an array of ρ^--mixing centered random variables with E Y n i 2 = a n i 2 E X i 2 < ∞ and b _ni = ±1 for i = 1, 2,..., n and σ n 2 = V a r ∑ i = 1 n b n i Y n i = 1. Note that X n 2 is an uniformly integrable family, we have

and

and ∀ ε > 0, we get

thus the conclusion is proved by Lemma 2.4.

Lemma 2.6 [2] Suppose that f₁(x) and f₂(y) are real, bounded, absolutely continuous functions on R with f ′ 1 ( x ) ≤ C 1 and f ′ 2 ( y ) ≤ C 2. Then for any random variables X and Y,

where X 2 , 1 = ∫ 0 ∞ P 1 2 X > x d x.

Lemma 2.7 Let {X _n , n ≥ 1} be a strictly stationary sequence of ρ^--mixing random variables with E X 1 = 0 , 0 < E X 1 2 < ∞ and

then for 0 < p < 2, we have

Proof By Lemma 2.1, we have

Let n _k = k ^α , where α > max 1 , p 2 - p. By (2.1), we get

From Borel-Cantelli lemma, it follows that

And by Lemma 2.2, it follows that

By Borel-Cantelli lemma, we conclude that

For every n, there exist n _k and n_k+1such that n _k ≤ n < n_k+1, by (2.2) and (2.3), we have

The proof is now completed.

Proof of Theorem 1.1 By the property of ρ^--mixing sequence, it is easy to see that {Y _n } is a strictly stationary ρ^--mixing sequence with EY₁ = 0 and E Y 1 2 = 1. We first prove

Let a n i = b i , n σ n , 1 ≤ i ≤ n , n ≥ 1. Obviously,

From condition (a₄) in Theorem 1.1 and Lemma 2.3, we have

and

By stationarity of {Y _n , n ≥ 1} and E |X₁|² < ∞, we know that Y n 2 is uniformly integrable, and from condition (a₂) in Theorem 1.1, we get ∑ n = 2 ∞ C o v Y 1 , Y n < ∞, so applying Lemma 2.5, we have

Notice that

so (3.1) is valid. Let f(x) be a bounded Lipschitz function and have a Radon-Nikodyn derivative h(x) bounded by Γ. From (3.1), we have

thus

On the other hand, note that (1.1) is equivalent to

from Section 2 of Peligrad and Shao [7] and Theorem 7.1 on P₄₂ from Billingsley [8]. Hence, to prove (3.3), it suffices to show that

by (3.2). Let ξ k = f S k , k σ k - E f S k , k σ k, 1 ≤ k ≤ n m we have

By the fact that f is bounded, we have

Now we estimate I₂, if l > 2k, we have

and

By Lemma 2.3 and condition (a₂) in Theorem 1.1, we have

and

By Lemma 2.6, the definition of ρ^--mixing sequence and condition (a₄), we have

By the inequality X 2 , 1 ≤ r r - 2 X r ( r > 2 ) (cf. Zhang [[2], p. 254] or Ledoux and Talagrand [[9], p. 251]), we get

and

thus

similarly,

hence

Similarly to (3.7), we have

and

Since f is a bounded Lipschitz function, we have