Abstract
Let {X n , n ≥ 1} be a strictly stationary ρ--mixing sequence of positive random variables with EX1 = μ > 0 and Var(X1) = σ2 < ∞. Denote and the coefficient of variation. Under suitable conditions, by the central limit theorem of weighted sums and the moment inequality we show that
where with is the distribution function of the random variable , and is a standard normal random variable.
MR(2000) Subject Classification: 60F15.
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1 Introduction and main results
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 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 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 .
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 , 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 with bk,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 and , where . Let , 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 and the coefficient of variation. Assume that
(a1) ,
(a2) ,
(a3) ρ-(n) = O(log-δn), ∃ δ > 1,
(a4) .
Then
Here and in the sequel, I{·} denotes indicator function and Φ(·) is the distribution function of standard normal random variable .
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 .
2 Some lemmas
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 , and
-
(i)
,
-
(ii)
,
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] .
Lemma 2.4 [[3], Theorem 3.2] Let {X ni , 1 ≤ i ≤ n, n ≥ 1} be an array of centered random variables with 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 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 be an array of real numbers such that and as n → ∞. If and is an uniformly integrable family, then
Proof Notice that
where 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 and b ni = ±1 for i = 1, 2,..., n and . Note that 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 and . Then for any random variables X and Y,
where .
Lemma 2.7 Let {X n , n ≥ 1} be a strictly stationary sequence of ρ--mixing random variables with and
then for 0 < p < 2, we have
Proof By Lemma 2.1, we have
Let n k = kα, where . 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 nk+1such that n k ≤ n < nk+1, by (2.2) and (2.3), we have
The proof is now completed.
3 Proof of the theorems
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 . We first prove
Let . 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 is uniformly integrable, and from condition (a2) in Theorem 1.1, we get , 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 , 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 (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
where . Hence if l > 2k, we have
Thus
Associated with (3.5), (3.6), and (3.8), we have
To prove (3.4), let , where τ > 1. From (3.9), we have
Thus ∀ε > 0, we have
By Borel-Cantelli lemma, we have
Note that
For every n, there exist n k and nk+1satisfying n k < n ≤ nk+1, we have
(3.4) is completed, so the proof of Theorem 1.1 is completed.
Proof of Theorem 1.2 Let , we have
Hence (1.1) is equivalent to
On the other hand, to prove (1.2), it suffices to show that
By Lemma 2.7, for enough large i, for some we have
It is easy to know that log(1+ x) = x + O(x2) for , thus
and
Hence for arbitrary small ε > 0, there is n0 = n0(ω, ε), such that for every n > n0 and arbitrary x,
so by (3.10), we know that (3.11) is true, and (3.11) is equivalent to (1.2), thus the proof of Theorem 1.2 is complete.
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Acknowledgements
The authors were very grateful to the editor and anonymous referees for their careful reading of the manuscript and valuable suggestions which helped in significantly improving an earlier version of this article. The study was supported by the National Natural Science Foundation of China (10926169, 11171003, 11101180), Key Project of Chinese Ministry of Education (211039), Foundation of Jilin Educational Committee of China (2012-158), and Basic Research Foundation of Jilin University (201001002, 201103204).
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XT and YZ carried out the design of the study and performed the analysis. YZ participated in its design and coordination. All authors read and approved the final manuscript.
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Tan, X., Zhang, Y. & Zhang, Y. An almost sure central limit theorem of products of partial sums for ρ--mixing sequences. J Inequal Appl 2012, 51 (2012). https://doi.org/10.1186/1029-242X-2012-51
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DOI: https://doi.org/10.1186/1029-242X-2012-51