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

Xili Tan; Ying Zhang; Yong Zhang

doi:10.1186/1029-242X-2012-51

Research
Open Access

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

Xili Tan¹Email author,
Ying Zhang¹ and
Yong Zhang²

Journal of Inequalities and Applications20122012:51

https://doi.org/10.1186/1029-242X-2012-51

Received: 13 November 2011
Accepted: 1 March 2012
Published: 1 March 2012

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 $S_{n} = \sum_{i = 1}^{n} X_{i}$ and $γ = \frac{σ}{μ}$ the coefficient of variation. Under suitable conditions, by the central limit theorem of weighted sums and the moment inequality we show that

\forall x = lim_{n \to \infty} \frac{1}{log n} \sum_{k = 1}^{n} \frac{1}{k} I \{{(\prod_{i = 1}^{k} \frac{S_{i}}{i μ})}^{\frac{1}{γ σ k}}\} = F (x) a . s .,

where $σ_{k}^{2} = V a r (S_{k, k}), S_{k, k} = \sum_{i = 1}^{k} b_{i, k} Y_{i}, b_{i, k} = \sum_{j = i}^{k} \frac{1}{j}, i \leq k$ with $b_{i, k} = 0, i > k, Y_{i} = \frac{X_{i - μ}}{σ}, F (x)$ is the distribution function of the random variable $e^{\sqrt{2} N}$ , and $N$ is a standard normal random variable.

MR(2000) Subject Classification: 60F15.

Keywords

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

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

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

ρ^{-} (X, Y) = 0 \lor sup \frac{C o v (f (X), g (Y))}{{(V a r f (X))}^{\frac{1}{2}} {(V a r g (Y))}^{\frac{1}{2}}},

where the sup is taken over all $f, g \in C$ 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,

C o v \{f (X_{i}, i \in S), g (X_{j}, j \in T)\} \leq 0,

whenever $f, g \in C$ .

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

ρ * (s) = sup \{ρ (S, T); S, T \subset N, dist (S, T) \geq s\} \to 0 a s s \to \infty,

where

ρ (S, T) = sup \{|E (f - E f) (g - E g) / ({∥f - E f∥}_{2} \cdot {∥g - E g∥}_{2})|; f \in L_{2} (σ (S)), g \in L_{2} (σ (T))\} .

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

ρ^{_} (s) = sup \{ρ^{_} (S, T); S, T \subset N, dist (S, T) \geq s\} \to 0 a s s \to \infty .

where,

ρ^{_} (S, T) = 0 \lor sup {\frac{C o v \{f (X_{i}, i \in S), g (X_{i}, j \in T)\}}{\sqrt{V a r \{f (X_{i}, i \in S)\} V a r \{g (X_{i}, j \in T)\}}}; f, g \in C} .

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 $ε_{0} > 0, V a r (\sum_{i = 1}^{n} \frac{1}{i} f (\frac{S_{i}}{σ_{i}})) = O ({log}^{2 - ε_{0}} n)$ , 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 $b_{k, n} = \sum_{i = k}^{n} \frac{1}{i}, k \leq n$ 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 ${\tilde{S}}_{n} = \sum_{k = 1}^{n} Y_{k}$ and $S_{n, n} = \sum_{k = 1}^{n} b_{k, n} Y_{k}$ , where $Y_{k} = \frac{X_{k} - μ}{σ}, k \geq 1$ . 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 $S_{n} = \sum_{i = 1}^{n} X_{i}$ and $γ = \frac{σ}{μ}$ the coefficient of variation. Assume that

(a₁) $σ_{1}^{2} = E X_{1}^{2} + 2 \sum_{n = 2}^{\infty} C o v (X_{1}, X_{n}) > 0$ ,

(a₂) $\sum_{n = 2}^{\infty} |C o v (X_{1}, X_{n})| < \infty$ ,

(a₃) ρ^-(n) = O(log^-δn), ∃ δ > 1,

(a₄) $inf_{n \in N} \frac{σ_{n}^{2}}{n} > 0$ .

Then

\forall x lim_{n \to \infty} \frac{1}{log n} \sum_{k = 1}^{n} \frac{1}{k} I \{\frac{S_{k, k}}{σ_{k}} \leq x\} = Φ (x) a . s .

(1.1)

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

\forall x = lim_{n \to \infty} \frac{1}{log n} \sum_{k = 1}^{n} \frac{1}{k} I \{{(\prod_{i = 1}^{k} \frac{S_{i}}{i μ})}^{\frac{1}{γ σ k}} \leq x\} = F (x) a . s .

(1.2)

Here and in the sequel, F(·) is the distribution function of the random variable $e^{\sqrt{2} N}$ .

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

E X_{n} = 0, 0 < E X_{1}^{2} < \infty

, and

(i)

$σ_{1}^{2} = E X_{1}^{2} + 2 \sum_{n = 2}^{\infty} C o v (X_{1}, X_{n}) > 0$ ,
(ii)

$\sum_{n = 2}^{\infty} |C o v (X_{1}, X_{n})| < \infty$ ,

then

\frac{E S_{n}^{2}}{n} \to {σ_{1}}^{2}, \frac{S_{n}}{σ_{1} \sqrt{n}} \overset{d}{\to} N (0, 1) a s n \to \infty .

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

E (max_{1 \leq j \leq n} {|S_{j}|}^{q}) \leq C \{\sum_{i = 1}^{n} E {|X_{i}|}^{q} + {(\sum_{i = 1}^{n} E X_{i}^{2})}^{\frac{q}{2}}\} .

Lemma 2.3 [6] $\sum_{i = 1}^{n} b_{i, n}^{2} = 2 n - b_{1, n}$ .

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} < \infty

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 (\sum_{i = 1}^{n} a_{n i} X_{n i})

and suppose that

sup_{n} \frac{1}{σ_{n}^{2}} \sum_{i = 1}^{n} E X_{n i}^{2} < \infty,

and

\underset{n \to \infty}{\lim \sup} \frac{1}{σ_{n}^{2}} \sum_{\underset{1 \leq i, j \leq n}{i, j : | i - j | \geq A}} C o v {(X_{n i}, X_{n j})}^{-} \to 0 a s A \to \infty,

and the following Lindeberg condition is satisfied:

\frac{1}{σ_{n}^{2}} \sum_{i = 1}^{n} E X_{n i}^{2} I \{|X_{n i}| \geq ε σ_{n}\} \to 0 a s n \to \infty

for every ε > 0. Then

\frac{1}{σ_{n}} \sum_{i = 1}^{n} a_{n i} X_{n i} \overset{d}{\to} N (0, 1) a s n \to \infty .

Lemma 2.5 Let {X_n, n ≥ 1} be a strictly stationary sequence of ρ^--mixing random variables with EX_n= 0 and

\sum_{n = 2}^{\infty} |C o v (X_{1}, X_{n})| < \infty, \{a_{n i}, 1 \leq i \leq n, n \geq 1\}

be an array of real numbers such that

sup_{n} \sum_{i = 1}^{n} a_{n i}^{2} < \infty

and

max_{1 \leq i \leq n} |a_{n i}| \to 0

as n → ∞. If

V a r (\sum_{i = 1}^{n} a_{n i} X_{i}) = 1

and

\{X_{n}^{2}\}

is an uniformly integrable family, then

\sum_{i = 1}^{n} a_{n i} X_{i} \overset{d}{\to} N (0, 1) a s n \to \infty .

Proof Notice that

\sum_{i = 1}^{n} a_{n i} X_{i} = \sum_{i = 1}^{n} \frac{a_{n i}}{|a_{n i}|} |a_{n i}| X_{i} = : \sum_{i = 1}^{n} b_{n i} Y_{n i},

where

b_{n i} = \frac{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} < \infty

and b_ni= ±1 for i = 1, 2,..., n and

σ_{n}^{2} = V a r (\sum_{i = 1}^{n} b_{n i} Y_{n i}) = 1

. Note that

\{X_{n}^{2}\}

is an uniformly integrable family, we have

sup_{n} \frac{1}{σ_{n}^{2}} \sum_{i = 1}^{n} E Y_{n i}^{2} = sup_{n} \sum_{i = 1}^{n} a_{n i}^{2} E X_{i}^{2} \leq sup_{n} \sum_{i = 1}^{n} a_{n i}^{2} \cdot sup_{i} E X_{i}^{2} < \infty,

and

\begin{array}{l} \underset{n \to \infty}{lim sup} \frac{1}{σ_{n}^{2}} \sum_{\underset{1 \leq i, j \leq n}{i, j : | i - j | \geq A}} C o v {(Y_{n i}, Y_{n j})}^{-} \\ = \underset{n \to \infty}{\lim \sup} \sum_{\underset{1 \leq i, j \leq n}{i, j : | i - j | \geq A}} C o v {(| a_{n i} | X_{i}, | a_{n j} | X_{j})}^{-} \\ \leq \underset{n \to \infty}{\lim \sup} \sum_{\underset{1 \leq i, j \leq n}{i, j : | i - j | \geq A}} | a_{n i} | \cdot | a_{n j} | \cdot | C o v (X_{i}, X_{j}) | \\ \leq C (\underset{n \to \infty}{\lim \sup} (\sum_{\underset{1 \leq i, j \leq n}{i, j : | i - j | \geq A}} {| a_{n i} |}^{2} \cdot | C o v (X_{i}, X_{j}) | + \sum_{\underset{1 \leq i, j \leq n}{i, j : | i - j | \geq A}} {| a_{n i} |}^{2} \cdot | C o v (X_{i}, X_{j}) |)) \\ \leq C \sup_{n} \sum_{i = 1}^{n} {| a_{n i} |}^{2} \cdot \sum_{i > A} | C o v (X_{1}, X_{i}) | \\ \to 0 a s A \to \infty, \end{array}

and ∀ ε > 0, we get

\begin{align} \frac{1}{σ_{n}^{2}} \sum_{i = 1}^{n} E Y_{n i}^{2} I \{|Y_{n i}| \geq ε σ_{n}\} \\ = \sum_{i = 1}^{n} a_{n i}^{2} E X_{i}^{2} I \{|a_{n i}| \cdot |X_{i}| \geq ε\} \\ \leq sup_{n} \sum_{i = 1}^{n} a_{n i}^{2} \cdot E X_{1}^{2} I \{|a_{n i}| \cdot |X_{1}| \geq ε\} \\ \leq sup_{n} \sum_{i = 1}^{n} a_{n i}^{2} \cdot E X_{1}^{2} I \{max_{1 \leq i \leq n} |a_{n i}| \cdot |X_{1}| \geq ε\} \to 0 a s n \to \infty, \end{align}

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)| \leq C_{1}

and

|{f^{'}}_{2} (y)| \leq C_{2}

. Then for any random variables X and Y,

|C o v (f_{1} (X), f_{2} (Y))| \leq C_{1} C_{2} \{- C o v (X, Y) + 8 p^{-} (X, Y) {∥X∥}_{2, 1} {∥Y∥}_{2, 1}\},

where ${∥X∥}_{2, 1} = \int_{0}^{\infty} P^{\frac{1}{2}} (|X| > x) d x$ .

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} < \infty

and

\begin{align} 0 < σ_{1}^{2} & = E X_{1}^{2} + 2 \sum_{n = 2}^{\infty} C o v (X_{1}, X_{n}) < \infty, \\ \sum_{n = 2}^{\infty} |C o v (X_{1}, X_{n})| < \infty, \end{align}

then for 0 < p < 2, we have

n^{- \frac{1}{p}} S_{n} \to 0 a . s . a s n \to \infty .

Proof By Lemma 2.1, we have

lim_{n \to \infty} \frac{E S_{n}^{2}}{n} = σ_{1}^{2} .

(2.1)

Let n_k= k^α, where

α > max \{1, \frac{p}{2 - p}\}

. By (2.1), we get

\sum_{k = 1}^{\infty} P \{|S_{n k}| \geq ε n_{k}^{\frac{1}{p}}\} \leq \sum_{k = 1}^{\infty} \frac{E S_{n_{k}}^{2}}{ε^{2} n_{k}^{\frac{2}{p}}} \leq \sum_{k = 1}^{\infty} \frac{C}{ε^{2} k^{α (\frac{2}{p} - 1)}} < \infty .

From Borel-Cantelli lemma, it follows that

n_{k}^{- \frac{1}{p}} S_{n_{k}} \to 0 a . s . a s k \to \infty .

(2.2)

And by Lemma 2.2, it follows that

\begin{align} \sum_{k = 1}^{\infty} P \{max_{n_{k} \leq n < n_{k + 1}} \frac{|S_{n} - S_{n_{k}}|}{n^{\frac{1}{p}}} \geq ε\} \leq \sum_{k = 1}^{\infty} \frac{E max_{n_{k} \leq n < n_{k + 1}} {|S_{n} - S_{n_{k}}|}^{2}}{ε^{2} n_{k}^{\frac{2}{p}}} \\ = \sum_{k = 1}^{\infty} \frac{E max_{n_{k} \leq n < n_{k + 1}} {|\sum_{i = n_{k + 1}}^{n} X_{i}|}^{2}}{ε^{2} n_{k}^{\frac{2}{p}}} \leq C \sum_{k = 1}^{\infty} \frac{(n_{k + 1} - n_{k})}{ε^{2} n_{k}^{\frac{2}{p}}} \leq C \sum_{k = 1}^{\infty} \frac{1}{k^{α (\frac{2}{p} - 1)}} < \infty . \end{align}

By Borel-Cantelli lemma, we conclude that

max_{n_{k} \leq n < n_{k + 1}} \frac{|S_{n} - S_{n_{k}}|}{n^{\frac{1}{p}}} \to 0 a . s . a s n \to \infty .

(2.3)

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

\begin{align} \frac{|S_{n}|}{n^{\frac{1}{p}}} & = \frac{|S_{n} - S_{n_{k}} + S_{n_{k}}|}{n^{\frac{1}{p}}} \\ \leq \frac{|S_{n_{k}}|}{n_{k}^{\frac{1}{p}}} + max_{n_{k} \leq n < n_{k + 1}} \frac{|S_{n} - S_{n_{k}}|}{n^{\frac{1}{p}}} \to 0 a . s . a s n \to \infty . \end{align}

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

E Y_{1}^{2} = 1

. We first prove

\frac{S_{n, n}}{σ_{n}} \overset{d}{\to} N (0, 1) a s n \to \infty .

(3.1)

Let

a_{n i} = \frac{b_{i, n}}{σ_{n}}, 1 \leq i \leq n, n \geq 1

. Obviously,

V a r (\sum_{i = 1}^{n} a_{n i} Y_{i}) = 1 .

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

sup_{n} \sum_{i = 1}^{n} a_{n i}^{2} = sup_{n} \sum_{i = 1}^{n} \frac{b_{i, n}^{2}}{σ_{n}^{2}} = sup_{n} \frac{2 n - b_{1, n}}{σ_{n}^{2}} \leq C sup_{n} \frac{2 n - b_{1, n}}{n} < \infty,

and

max_{1 \leq i \leq n} |a_{n i}| = max_{1 \leq i \leq n} \frac{b_{i, n}}{σ_{n}} \leq \frac{b_{1, n}}{σ_{n}} \leq \frac{C log n}{\sqrt{n}} \to 0 a s n \to \infty .

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

\sum_{n = 2}^{\infty} |C o v (Y_{1}, Y_{n})| < \infty

, so applying Lemma 2.5, we have

\sum_{i = 1}^{n} a_{n i} Y_{i} \overset{d}{\to} N (0, 1) .

Notice that

\sum_{i = 1}^{n} a_{n i} Y_{i} = \sum_{i = 1}^{n} \frac{b_{i, n} Y_{i}}{σ_{n}} = \frac{S_{n, n}}{σ_{n}},

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

E f (\frac{S_{n, n}}{σ_{n}}) \to E f (N (0, 1)) a s n \to \infty,

thus

\frac{1}{log n} \sum_{k = 1}^{n} \frac{1}{k} E f (\frac{S_{k, k}}{σ_{k}}) - E f (N (0, 1)) \to 0 a s n \to \infty .

(3.2)

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

lim_{n \to \infty} \frac{1}{log n} \sum_{k = 1}^{n} \frac{1}{k} f (\frac{S_{k, k}}{σ_{k}}) = \int_{- \infty}^{\infty} f (x) d Φ (x) = E f (N (0, 1)) a . s .

(3.3)

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

T_{n} = \frac{1}{log n} \sum_{k = 1}^{n} \frac{1}{k} [f (\frac{S_{k, k}}{σ_{k}}) - E f (\frac{S_{k, k}}{σ_{k}})] \to 0 a . s . n \to \infty

(3.4)

by (3.2). Let

ξ_{k} = f (\frac{S_{k, k}}{σ_{k}}) - E f (\frac{S_{k, k}}{σ_{k}})

, 1 ≤ k ≤ n m we have

\begin{align} E T_{n}^{2} & = \frac{1}{{log}^{2} n} E {(\sum_{k = 1}^{n} \frac{ξ_{k}}{k})}^{2} \\ \leq \frac{1}{{log}^{2} n} \sum_{1 \leq k \leq l \leq n, 2 k \geq l} \frac{|E ξ_{k} ξ_{l}|}{k l} + \frac{1}{{log}^{2} n} \sum_{1 \leq k \leq l \leq n, 2 k \geq l} \frac{|E ξ_{k} ξ_{l}|}{k l} \\ : = I_{1} + I_{2} . \end{align}

(3.5)

By the fact that f is bounded, we have

I_{1} \leq \frac{C}{{log}^{2} n} \sum_{k = 1}^{n} \sum_{l = k}^{2 k} \frac{1}{k l} = \frac{C}{{log}^{2} n} \sum_{k = 1}^{n} \frac{1}{k} \sum_{l = k}^{2 k} \frac{1}{l} \leq C ({log}^{- 1} n) .

(3.6)

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

\begin{align} S_{l, l} - S_{2 k, 2 k} & = (b_{1, l} Y_{1} + b_{2, l} Y_{2} + \dots + b_{l, l} Y_{l}) - (b_{1, 2 k} Y_{1} + b_{2, 2 k} Y_{2} + \dots + b_{2 k, 2 k} Y_{2 k}) \\ = (b_{2 k + 1, l} Y_{2 k + 1} + \dots + b_{l, l} Y_{l}) + b_{2 k + 1, l} {\tilde{S}}_{2 k}, \end{align}

and

\begin{align} |E ξ_{k} ξ_{l}| & = |C o v (f (\frac{S_{k, k}}{σ_{k}}), f (\frac{S_{l, l}}{σ_{l}}))| \\ \leq |C o v (f (\frac{S_{k, k}}{σ_{k}}), f (\frac{S_{l, l}}{σ_{l}}) - f (\frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}))| \\ + |C o v (f (\frac{S_{k, k}}{σ_{k}}), f (\frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}))| . \end{align}

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

\begin{align} V a r (S_{k, k}) & = \sum_{i = 1}^{k} b_{i, k}^{2} E Y_{i}^{2} + 2 \sum_{j = 1}^{k - 1} \sum_{i = j + 1}^{k} b_{i, k} b_{j, k} C o v (Y_{i}, Y_{j}) \\ \leq \sum_{i = 1}^{k} b_{i, k}^{2} + 2 \sum_{j = 1}^{k} b_{j, k}^{2} \sum_{i = j + 1}^{k} |C o v (Y_{i}, Y_{j})| \\ \leq C k, \end{align}

(3.7)

and

\begin{align} V a r (S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}) \\ = \sum_{i = 2 k + 1}^{l} b_{i, l}^{2} E Y_{i}^{2} + 2 \sum_{j = 2 k + 1}^{l - 1} \sum_{i = j + 1}^{l} b_{i, l} b_{j, l} C o v (Y_{i}, Y_{j}) \\ \leq \sum_{i = 2 k + 1}^{l} b_{i, l}^{2} + 2 \sum_{j = 1}^{l} b_{i, l}^{2} \sum_{i = j + 1}^{l} |C o v (Y_{i}, Y_{j})| \\ \leq C l . \end{align}

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

\begin{align} |C o v (f (\frac{S_{k, k}}{σ_{k}}), f (\frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}))| \\ \leq C \{- C o v (\frac{S_{k, k}}{σ_{k}}, \frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}) \\ + 8 ρ^{-} (\frac{S_{k, k}}{σ_{k}}, \frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}) \cdot {∥\frac{S_{k, k}}{σ_{k}}∥}_{2, 1} \cdot {∥\frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}∥}_{2, 1}\} \\ \leq C ρ^{-} (k) {(V a r (\frac{S_{k, k}}{σ_{k}}))}^{\frac{1}{2}} {(V a r (\frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}))}^{\frac{1}{2}} \\ + C ρ^{-} (k) {∥\frac{S_{k, k}}{σ_{k}}∥}_{2, 1} \cdot {∥\frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}∥}_{2, 1} \\ \leq C ρ^{-} (k) + C ρ^{-} (k) {∥\frac{S_{k, k}}{σ_{k}}∥}_{2, 1} \cdot {∥\frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}∥}_{2, 1} . \end{align}

By the inequality

{∥X∥}_{2, 1} \leq \frac{r}{r - 2} {∥X∥}_{r} (r > 2)

(cf. Zhang [[2], p. 254] or Ledoux and Talagrand [[9], p. 251]), we get

{∥\frac{S_{k, k}}{σ_{k}}∥}_{2, 1} \leq \frac{r}{r - 2} {∥\frac{S_{k, k}}{σ_{k}}∥}_{r} = \frac{r}{r - 2} \frac{1}{σ_{k}} {(E {|S_{k, k}|}^{r})}^{\frac{1}{r}},

and

\begin{align} E {|S_{k, k}|}^{r} & = E | \sum_{j = 1}^{k} b_{j, k} Y_{j} |^{r} \\ \leq C \{\sum_{j = 1}^{k} b_{j, k}^{r} E {|X_{j}|}^{r} + {(\sum_{j = 1}^{k} b_{j, k}^{2} E X_{j}^{2})}^{\frac{r}{2}}\} \\ \leq C \{k ({log}^{r} k) + k^{\frac{r}{2}}\}, \end{align}

thus

{∥\frac{S_{k, k}}{σ_{k}}∥}_{2, 1} \leq C (\frac{r}{r - 2} \cdot \frac{log k}{k^{\frac{1}{2} - \frac{1}{r}}} + \frac{r}{r - 2}) < C,

similarly,

{∥\frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}∥}_{2, 1} < C,

hence

|C o v (f (\frac{S_{k, k}}{σ_{k}}), f (\frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}))| \leq C ρ^{-} (k) .

Similarly to (3.7), we have

\begin{align} V a r (S_{2 k, 2 k}) & = \sum_{i = 1}^{2 k} b_{i, 2 k}^{2} E Y_{i}^{2} + 2 \sum_{j = 1}^{2 k - 1} \sum_{i = j + 1}^{2 k} b_{i, 2 k} b_{j, 2 k} C o v (Y_{i}, Y_{j}) \\ \leq \sum_{i = 1}^{2 k} b_{i, 2 k}^{2} + 2 \sum_{j = 1}^{2 k - 1} b_{i, 2 k}^{2} \sum_{i = j + 1}^{2 k} |C o v (Y_{i}, Y_{j})| \\ \leq C k, \end{align}

and

\begin{align} V a r ({\tilde{S}}_{2 k}) & = V a r (\sum_{i = 1}^{2 k} Y_{i}) \\ = \sum_{i = 1}^{2 k} E Y_{i}^{2} + 2 \sum_{i = 1}^{2 k - 1} \sum_{j = i + 1}^{2 k} |C o v (Y_{i}, Y_{j})| \\ = 2 k + 2 \sum_{i = 1}^{2 k - 1} \sum_{j = 2}^{2 k - i + 1} C o v (Y_{i}, Y_{j}) \\ \leq C k . \end{align}

Since f is a bounded Lipschitz function, we have

\begin{align} |C o v (f (\frac{S_{k, k}}{σ_{k}}), f (\frac{S_{l, l}}{σ_{l}}) - f (\frac{S_{l, l} - S_{2 k, 2 k} - b_{2 k + 1, l} {\tilde{S}}_{2 k}}{σ_{l}}))| \\ \leq C E \frac{|S_{2 k, 2 k} + b_{2 k + 1, l} {\tilde{S}}_{2 k}|}{σ_{l}} \\ \leq C \frac{{(V a r (S_{2 k, 2 k}))}^{\frac{1}{2}}}{σ_{l}} + C \frac{b_{2 k + 1, l} {(V a r ({\tilde{S}}_{2 k}))}^{\frac{1}{2}}}{σ_{l}} \\ \leq C {(\frac{k}{l})}^{\frac{1}{2}} + C {(\frac{k}{l})}^{\frac{1}{2}} log \frac{1}{2 k} \\ \leq C {(\frac{k}{l})}^{ε}, \end{align}

where

0 < ε < \frac{1}{2}

. Hence if l > 2k, we have

|E ξ_{k} ξ_{l}| \leq C [ρ^{-} (k) + {(\frac{k}{l})}^{ε}] .

Thus

\begin{align} I_{2} & \leq \frac{C}{{log}^{2} n} \sum_{l = 2}^{n} \sum_{k = 1}^{l - 1} \frac{1}{k^{1 - ε} l^{1 + ε}} + \frac{C}{{log}^{2} n} \sum_{l = 2}^{n} \frac{1}{l} \sum_{k = 1}^{l - 1} \frac{ρ^{-} (k)}{k} \\ \leq \frac{C}{{log}^{2} n} \sum_{l = 2}^{n} \frac{1}{l^{1 + ε}} \frac{{(l - 1)}^{ε}}{ε} + \frac{C}{{log}^{2} n} \sum_{l = 2}^{n} \frac{1}{l} \sum_{k = 1}^{n} \frac{{log}^{- δ} k}{k} \\ \leq \frac{C}{{log}^{2} n} \sum_{l = 2}^{n} \frac{1}{l} + \frac{C}{{log}^{2} n} \sum_{l = 2}^{n} \frac{1}{l} \sum_{k = 1}^{n} \frac{{log}^{- δ} k}{k} \\ \leq C {log}^{- 1} n . \end{align}

(3.8)

Associated with (3.5), (3.6), and (3.8), we have

E T_{n}^{2} \leq C {log}^{- 1} n .

(3.9)

To prove (3.4), let

n_{k} = e^{k^{τ}}

, where τ > 1. From (3.9), we have

\sum_{k = 1}^{\infty} E T_{n_{k}}^{2} \leq C \sum_{k = 1}^{\infty} {log}^{- 1} n_{k} = C \sum_{k = 1}^{\infty} \frac{1}{k^{τ}} < \infty .

Thus ∀ε > 0, we have

\sum_{k = 1}^{\infty} P \{|T_{n_{k}}| \geq ε\} \leq \sum_{k = 1}^{\infty} \frac{E T_{n_{k}}^{2}}{ε^{2}} < \infty .

By Borel-Cantelli lemma, we have

T_{n_{k} \to 0} a . s . a s k \to \infty .

Note that

\frac{log n_{k + 1}}{log n_{k}} = \frac{{(k + 1)}^{τ}}{k^{τ}} \to 1 a s k \to \infty .

For every n, there exist n_kand n_k+1satisfying n_k< n ≤ n_k+1, we have

\begin{align} |T_{n}| & \leq \frac{1}{log n_{k}} |\sum_{i = 1}^{n_{k}} \frac{ξ_{i}}{i}| + \frac{1}{log n_{k}} \sum_{i = n_{k}}^{n_{k + 1}} \frac{|ξ_{i}|}{i} \\ \leq |T_{n_{k}}| + C (\frac{log n_{k + 1}}{log n_{k}} - 1) \to 0 a . s . a s n \to \infty, \end{align}

(3.4) is completed, so the proof of Theorem 1.1 is completed.

Proof of Theorem 1.2 Let

C_{i} = \frac{S_{i}}{μ_{i}}

, we have

\frac{1}{γ σ_{k}} \sum_{i = 1}^{k} (C_{i} - 1) = \frac{1}{γ σ_{k}} \sum_{i = 1}^{k} (\frac{S_{i}}{μ i} - 1) = \frac{1}{σ_{k}} \sum_{i = 1}^{k} b_{i, k} Y_{i} = \frac{S_{k, k}}{σ_{k}} .

Hence (1.1) is equivalent to

\forall x lim_{n \to \infty} \frac{1}{log n} \sum_{k = 1}^{n} \frac{1}{k} I \{\frac{1}{γ σ_{k}} \sum_{i = 1}^{k} (C_{i} - 1) \leq x\} = Φ (x) a . s .

(3.10)

On the other hand, to prove (1.2), it suffices to show that

\forall x lim_{n \to \infty} \frac{1}{log n} \sum_{k = 1}^{n} \frac{1}{k} I \{\frac{1}{γ σ_{k}} \sum_{i = 1}^{k} log C_{i} \leq x\} = Φ (x) a . s .

(3.11)

By Lemma 2.7, for enough large i, for some

\frac{4}{3} < p < 2

we have

|C_{i} - 1| = |\frac{S_{i}}{μ i} - 1| \leq C i^{\frac{1}{p} - 1} a . s .

It is easy to know that log(1+ x) = x + O(x²) for

|x| < \frac{1}{2}

, thus

|\sum_{k = 1}^{n} log C_{k} - \sum_{k = 1}^{n} (C_{k} - 1)| \leq C \sum_{k = 1}^{n} {(C_{k} - 1)}^{2} \leq C \sum_{k = 1}^{n} k^{\frac{2}{p} - 2} \leq C n^{\frac{2}{p} - 1} a . s .,

and

\sum_{k = 1}^{n} (C_{k} - 1) - C n^{\frac{2}{p} - 1} \leq \sum_{k = 1}^{n} log C_{k} \leq \sum_{k = 1}^{n} (C_{k} - 1) + C n^{\frac{2}{p} - 1} a . s .

Hence for arbitrary small ε > 0, there is n₀ = n₀(ω, ε), such that for every n > n₀ and arbitrary x,

I \{\frac{1}{γ σ_{k}} \sum_{i = 1}^{k} (C_{i} - 1) \leq x - ε\} \leq I \{\frac{1}{γ σ_{k}} \sum_{i = 1}^{k} log C_{i} \leq x\} \leq I \{\frac{1}{γ σ_{k}} \sum_{i = 1}^{k} (C_{i} - 1) \leq x + ε\},

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.

Declarations

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).

Authors’ Affiliations

(1)

Department of Mathematics, Beihua University, Jilin, 132013, P.R. China

(2)

College of Mathematics, Jilin University, Changchun, 130012, P.R. China

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

Acknowledgements

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