A Note on the Central Limit Theorems for Dependent Random Variables

Classical central limit theorem is considered the heart of probability and statistics theory. Our interest in this paper is central limit theorems for functions of random variables under mixing conditions. We impose mixing conditions on the differences between the joint cumulative distribution functions and the product of the marginal cumulative distribution functions. By using characteristic functions, we obtain several limit theorems extending previous results.


Introduction
The central limit theorem is one of the most remarkable results of the theory of probability 1 , which is critical to understand inferential statistics and hypothesis testing 2, 3 .The assumption of independence for a sequence of observations X 1 , X 2 , . . . is often a technical convenience.Real data frequently exhibit some dependence and at the least some correlation at small lags.One extensively investigated kind of dependence is the m-dependence case see e.g., 4-8 , in which random variables are considered as independent as long as they are mstep apart.More general measures of dependence are called mixing conditions, which are derived from the estimation of the difference between characteristic functions of averages of dependent and independent random variables.These conditions have some appealing physical interpretations.Various mixing conditions have been proposed by researchers; see e.g., 9-14 to name just a few.
Our main interest in this note is the central limit theorem for dependent classes of random variables.Following the work 12 , instead of estimating the difference between the characteristic functions of the sum of dependent random variables and those of the sum of independent random variables of the same distributions, we compute the exact value of this difference.Our results weaken the conditions imposed on the random variable sequences in Theorem 1 and Theorem 2 of 12 and can be used to describe systems which are globally determined but locally random.It is noteworthy that the work 12 has been extended in another direction, where the sum of a random number of random variables is examined 15 .
The rest of the paper is organized as follows.In Section 2, we present our central limit theorems, and in Section 3, we give the proofs.

Main Results
Before proceeding, we introduce some notations.For a, b ∈ R, denote by a ∨ b and a ∧ b the maximal and minimal values between them, respectively.Let X 1 , X 2 , . . .and Z be random variables.The image of Z is denoted by I Z ⊆ R. We write X n D − → Z meaning that X n converges in distribution to Z as n → ∞.Denote by N 0, 1 the standard normal variable.Let E Z or simply EZ and Var Z represent the mean and variance of Z, respectively.Theorem 2.1.Let {X i } i≥1 be a sequence of identically distributed random variables and {f i } i≥1 a sequence of measurable functions such that f i : for some γ i > 0 and α i , β 2 ≤ 0, and holds, where v 1 , . . ., v j is any choice of indices such that x for all i, which is the special case considered in 12 Theorem 3 .It is worth noting that in the conditions of Theorem 2.1 we used the expression X γ 1 1 , where γ 1 is a real number.Naturally, we preclude the situation where the random variable X 1 takes negative values and γ 1 is not an integer, since such kind of power is undefined.
Fix k and let j be relatively small, compared to k, then the right-hand side of 2.3 is close to zero.If we let j be close to k − k ε 1 , then the right-hand side of 2.3 is close to 1. Hence, the dependence condition 2.3 allows a stronger dependence within a larger class of random variables and demands more independence within a smaller class.This structure can be used to describe systems which are globally determined but locally random.
Theorem 2.1 and Corollary 2.2 deal with dependence conditions on an "event-toevent" basis, while the following analogous results consider dependence on an "average" matter.
Theorem 2.3.Let {X i } i≥1 be a sequence of identically distributed random variables and {f i } i≥1 a sequence of measurable functions such that f i : or for some γ i > 0 and α i , β 2 ≤ 0, and where v 1 , . . ., v j is any choice of indices such that Corollary 2.4.Let {X i } i≥1 be a sequence of identically distributed random variables and {f i } i≥1 a sequence of measurable functions such that f i : Suppose there exists δ, 0 < δ < 1, such that for sufficiently large k the inequality holds, where v 1 , . . ., v j is any choice of indices such that as k → ∞.

Proofs
We will prove Theorems 2.1 and 2.3 in this section through several lemmas, and the proof of corollaries follows directly.For convenience, denote by the product of all a i 's except a v 1 , . . ., a v j .The following lemma regarding the difference between characteristic functions of sums of dependent and independent random variables is stated in 12 without proof.Lemma 3.1.Let X 1 , X 2 , . . ., X k be random variables satisfying 0 ≤ X i ≤ M, 1 ≤ i ≤ k.Let g 1 , . . ., g k : 0, M → C be absolutely continuous integrable functions.Then one has the identity

3.1
Proof.Let F i x P X i ≤ x for i ≥ 1.Therefore, for k 1, we have by integration by parts.Let F x 1 , x 2 P X 1 ≤ x 1 , X 2 ≤ x 2 .Then, for k 2, we derive from 3.2 that

3.3
The general results then follow by induction.
The following lemma compares with the Lindeberg condition.Lemma 3.2.Let {X i } i≥1 be a sequence of identically distributed random variables and f i :

ISRN Probability and Statistics
Proof.Without loss of generality, we can assume that f i X i are centered at 0, that is, Ef i X i 0. In fact, let f i : f i − Ef i X 1 , and then Ef i X i 0. By the assumptions 2.1 and 2.2 , we have Therefore, f i : β 1 ≤ 0. Hence, we may assume that Ef i X i 0 without loss of generality. Choose We will show that the constructed sequence {ε k } k≥1 is what we want.Since s 2 k k i 1 Var f i X i and X i 's are identically distributed, by virtue of the bounds in 2.1 and 2.2 we obtain β 2 2 .By our assumptions, we have ISRN Probability and Statistics 7 as k → ∞, which verifies 3.4 .Let Δ X 1 : k, the quantity before the integration in 3.11 tends to zero.Consequently, 3.5 readily holds since the integration in 3.11 is bounded using our assumptions.Now we want to verify 3.6 .Note that

3.12
By the Markovian inequality see e.g., 1 , where the first integration in 3.14 tends to EΔ X 1 2 < ∞ involving 3.13 .Therefore, the second integration in 3.14 tends to zero, which, together with 3.12 , finally verifies 3.6 .
Proof of Theorem 2.1.We assume that Ef i X i 0. For large enough k, we have In what follows, we suppose that k ε 1 is an integer for notation convenience.In the sequel, we take a specific sequence {ε k } k≥1 as in Lemma 3.2, and then 3.15 amounts to

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We define a sequence {Y k,j } 1≤j≤k−k ε 1 of random variables as follows:

3.16
Hence, we have 0 The inequality 2.3 then implies that, for large enough k, 3.17 Next, we calculate the difference between characteristic functions of sums of independent and dependent random variables.Taking where the second inequality comes from 3.17 and c is a positive constant.Since ε 1 < ε 2 and s k /c k → 1 as k → ∞, 3.18 tends to 0 as k → ∞.Define a sequence {W j } 1≤j≤k−k ε 1 of independent random variables such that W j has the same distribution with Y j .By our construction, for sufficiently large k,

3.19
Involving Lemma 3.2, it yields that Y k,j satisfies the Lindeberg condition.Accordingly, W j also satisfies the Lindeberg condition.Thus, the central limit theorem is true for W j since 3.18 approaches 0, which implies that the central limit theorem holds for Y k,j , that is,

3.21
On the other hand, it is obvious that

3.22
The first quantity on the right-hand side of 3.22 is up-bounded by as k → ∞, using 3.15 .Combining 3.5 , 3.22 , and 3.23 , we obtain

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Proof of Theorem 2.3.We define Y k,j , c k , and ε k as in the proof of Theorem 2.1.We obtain where c is a positive constant.Since s k Θ √ k and s k /c k → 1 as k → ∞, the expression 3.26 tends to 0 as k → ∞.The remaining part of the proof follows as in the proof of Theorem 2.1.
.20 as k → ∞.Taking into account that s k /c k → 1 and the definition of Y k,j , by 3.
concludes the proof of Theorem 2.1.
Let {X i } i≥1 be a sequence of identically distributed random variables and {f i } i≥1 a sequence of measurable functions such that f i : I X 1 → R satisfying, for all x ∈ I X 1 , expressions 2.1 or 2.2 .Assume that E|X 1 | n < ∞ for n ∈ N. Suppose there exists δ, 0 < δ < 1, such that for sufficiently large k the inequality