# A Note on the Central Limit Theorems for Dependent Random Variables

Authors: Yilun Shang
Journal: ISRN Probability and Statistics
 (2012)

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## Abstract

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.


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## Body

## 1.  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 X1,X2,… 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 m-step 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 Section2, we present our central limit theorems, and in Section 3, we give the proofs.


## 2.  Main Results

Before proceeding, we introduce some notations. Fora,b∈ℝ, denote by a∨b and a∧b the maximal and minimal values between them, respectively. Let X1,X2,… and Z be random variables. The image of Z is denoted by 𝕀(Z)⊆ℝ. We write Xn→DZ meaning that Xn 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{Xi}i≥1 be a sequence of identically distributed random variables and {fi}i≥1 a sequence of measurable functions such that fi:𝕀(X1)→ℝ satisfying, for all x∈𝕀(X1), either
(2.1)0≤α1xγ1+β1≤fi(x)≤α2xγ2+β2,
or
(2.2)α1xγ1+β1≤fi(x)≤α2xγ2+β2≤0,
for some γi>0 and αi,βi∈ℝ, i=1,2. Assume that α1EX1γ1+β1≥0, α2EX1γ2+β2≤0, and E|X1|(γ1∨γ2)(2+ε)<+∞ for some ε>0. Let 0<ε1<ε/(2(1+ε)) and sk2=∑i=1k‍Var(fi(Xi)). If for sufficiently large k(2.3)sup{|P(⋂i=1j{fvi(Xvi)≤xvi})-∏i=1j‍P(fvi(Xvi)≤xvi)|:(xv1,…,xvj)∈ℝj}≤(1-k-ε1)k-kε1-j,
where v1,…,vj is any choice of indices such that kε1<v1<⋯<vj≤k, then
(2.4)1sk∑i=1k‍(fi(Xi)-Efi(Xi))→DN(0,1)
as k→∞.Corollary 2.2.
Let{Xi}i≥1 be a sequence of identically distributed random variables and {fi}i≥1 a sequence of measurable functions such that fi:𝕀(X1)→ℝ satisfying, for all x∈𝕀(X1), expressions (2.1) or (2.2). Assume that E|X1|n<+∞ for n∈ℕ. Suppose there exists δ, 0<δ<1, such that for sufficiently large k the inequality
(2.5)sup{|P(⋂i=1j{fvi(Xvi)≤xvi})-∏i=1j‍P(fvi(Xvi)≤xvi)|:(xv1,…,xvj)∈ℝj}≤(1-k-δ/2)k-kδ/2-j
holds, where v1,…,vj is any choice of indices such that kδ/2<v1<⋯<vj≤k, then
(2.6)1sk∑i=1k‍(fi(Xi)-Efi(Xi))→DN(0,1)
as k→∞.Generally,{fi}i≥1 is a sequence of nonlinearly bounded functions. If we set α1=α2=γ1=γ2=1 and β1=β2=0, then fi(x)=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 X1γ1, where γ1 is a real number. Naturally, we preclude the situation where the random variable X1 takes negative values and γ1 is not an integer, since such kind of power is undefined.Fixk 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-kε1, 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.Theorem2.1 and Corollary 2.2 deal with dependence conditions on an “event-to-event” basis, while the following analogous results consider dependence on an “average” matter.Theorem 2.3.
Let{Xi}i≥1 be a sequence of identically distributed random variables and {fi}i≥1 a sequence of measurable functions such that fi:𝕀(X1)→ℝ satisfying, for all x∈𝕀(X1), either
(2.7)0≤α1xγ1+β1≤fi(x)≤α2xγ2+β2,
or
(2.8)α1xγ1+β1≤fi(x)≤α2xγ2+β2≤0,
for some γi>0 and αi,βi∈ℝ, i=1,2. Assume that α1EX1γ1+β1≥0, α2EX1γ2+β2≤0, and E|X1|(γ1∨γ2)(2+ε)<+∞ for some ε>0. Let 0<ε1<ε/(2(1+ε)) and sk2=∑i=1k‍Var(fi(Xi)). If for sufficiently large k(2.9)∫‍ℝj|P(⋂i=1j{fvi(Xvi)≤xvi})-∏i=1j‍P(fvi(Xvi)≤xvi)|dxv1⋯dxvj≤(1-k-ε1)k-kε1-j,
where v1,…,vj is any choice of indices such that kε1<v1<⋯<vj≤k, then
(2.10)1sk∑i=1k‍(fi(Xi)-Efi(Xi))→DN(0,1)
as k→∞.Corollary 2.4.
Let{Xi}i≥1 be a sequence of identically distributed random variables and {fi}i≥1 a sequence of measurable functions such that fi:𝕀(X1)→ℝ satisfying, for all x∈𝕀(X1), expressions (2.7) or (2.8). Assume that E|X1|n<+∞ for n∈ℕ. Suppose there exists δ, 0<δ<1, such that for sufficiently large k the inequality
(2.11)∫‍ℝj|P(⋂i=1j{fvi(Xvi)≤xvi})-∏i=1j‍P(fvi(Xvi)≤xvi)|dxv1⋯dxvj≤(1-k-δ/2)k-kδ/2-j
holds, where v1,…,vj is any choice of indices such that kδ/2<v1<⋯<vj≤k, then
(2.12)1sk∑i=1k‍(fi(Xi)-Efi(Xi))→DN(0,1)
as k→∞.


## 3. Proofs

We will prove Theorems2.1 and 2.3 in this section through several lemmas, and the proof of corollaries follows directly.For convenience, denote by(∏i=1k‍ai)/(av1⋯avj), 1≤v1<⋯<vj≤k, the product of all ai’s except av1,…,avj. 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.
LetX1,X2,…,Xk be random variables satisfying 0≤Xi≤M, 1≤i≤k. Let g1,…,gk:[0,M]→ℂ be absolutely continuous integrable functions. Then one has the identity
(3.1)E(∏i=1k‍gi(Xi))-∏i=1k‍Egi(Xi)=∑j=2k‍∑1≤v1<⋯<vj≤k‍(-1)j∏i=1k‍gi(M)gv1(M)⋯gvj(M)·∫0M‍⋯∫0M‍∏i=1j‍gvi′(xvi)×(P(⋂i=1j{Xvi≤xvi})-∏i=1j‍P(Xvi≤xvi))dxv1⋯dxvj.Proof.
LetFi(x)=P(Xi≤x) for i≥1. Therefore, for k=1, we have
(3.2)E(g1(X1))=∫0M‍g1(x)dF1(x)=g1(M)-∫0M‍F1(x)g1′(x)dx
by integration by parts. Let F(x1,x2)=P(X1≤x1,X2≤x2). Then, for k=2, we derive from (3.2) that
(3.3)E(g1(X1)g2(X2))-Eg1(X1)Eg2(X2)=∫∫0M‍g1′(x1)g2′(x2)F(x1,x2)dx1dx2-∫0M‍g1′(x1)F1(x1)dx1∫0M‍g2′(x2)F2(x2)dx2=∫∫0M‍g1′(x1)g2′(x2)·(F(x1,x2)-F1(x1)F2(x2))dx1dx2.
The general results then follow by induction.The following lemma compares with the Lindeberg condition.Lemma 3.2.
Let{Xi}i≥1 be a sequence of identically distributed random variables and fi:𝕀(X1)→ℝ satisfying, for all x∈𝕀(X1), expressions (2.1) or (2.2), i≥1. Assume that α1EX1γ1+β1≥0, α2EX1γ2+β2≤0, and E|X1|(γ1∨γ2)(2+ε)<+∞ for some ε>0. Let 0<ε1<ε/(2(1+ε)). Then, there exists a sequence of reals {εk}k≥1 such that
(3.4)εk→0,εksk→+∞ask→∞,(3.5)1sk∑i=1k‍∫{|fi(Xi)|>εksk}‍|fi(Xi)|dP→0ask→∞,(3.6)1sk2∑i=1k‍∫{|fi(Xi)|>εksk}‍fi2(Xi)dP→0ask→∞.Proof.
Without loss of generality, we can assume thatfi(Xi) are centered at 0, that is, Efi(Xi)=0. In fact, let fi′:=fi-Efi(X1), and then Efi′(Xi)=0. By the assumptions (2.1) and (2.2), we have
(3.7)-∞<α1EX1γ1+β1≤Efi(X1)≤α2EX1γ2+β2<+∞.
Therefore, fi′:𝕀(X1)→ℝ satisfies
(3.8)α1xγ1+β1-(α2EX1γ2+β2)≤fi′(x)≤α2xγ2+β2-(α1EX1γ1+β1).
For x∈𝕀(X1), if (2.1) holds, then α1xγ1+β1-(α2EX1γ2+β2)≥0; if (2.2) holds, then α2xγ2+β2-(α1EX1γ1+β1)≤0. Furthermore, by assumptions, we have α1EX1γ1+β1-(α2EX1γ2+β2)≥0 and α2EX1γ2+β2-(α1EX1γ1+β1)≤0. Hence, we may assume that Efi(Xi)=0 without loss of generality.
Chooseε2 so that ε1<ε2<ε/(2(1+ε)). Define εk=k-ε2 for k≥3. We will show that the constructed sequence {εk}k≥1 is what we want.
Sincesk2=∑i=1k‍Var(fi(Xi)) and Xi’s are identically distributed, by virtue of the bounds in (2.1) and (2.2) we obtain
(3.9)kC12≤sk2≤kC22,
where C12=E((α12X12γ1+2α1β1X1γ1+β12)∧(α22X12γ2+2α2β2X1γ2+β22)) and C22=E((α12X12γ1+2α1β1X1γ1+β12)∨(α22X12γ2+2α2β2X1γ2+β22)).
By our assumptions, we have(3.10)εksk≥kkε2C1→+∞
as k→∞, which verifies (3.4). Let Δ(X1):=(|α1||X1|γ1+|β1|)∨(|α2||X1|γ2+|β2|). We have
(3.11)1sk∑i=1k‍∫{|fi(Xi)|>εksk}‍|fi(Xi)|dP≤ksk∫{Δ(X1)>εksk}‍Δ(X1)dP≤ksk(1εksk)1+ε∫{Δ(X1)>εksk}‍Δ(X1)2+εdP.
Since ε2<ε/(2(1+ε)) and C1k≤sk≤C2k, 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)1sk2∑i=1k‍∫{|fi(Xi)|>εksk}‍fi2(Xi)dP≤ksk2∫{Δ(X1)>εksk}‍Δ(X1)2dP=O(∫{Δ(X1)>εksk}‍Δ(X1)2dP).
By the Markovian inequality (see e.g., [1]),
(3.13)P(Δ(X1)2>εk2sk2)≤EΔ(X1)2εk2sk2→0
as k→∞. We have
(3.14)EΔ(X1)2=∫{Δ(X1)2≤εk2sk2}‍Δ(X1)2dP+∫{Δ(X1)2>εk2sk2}‍Δ(X1)2dP,
where the first integration in (3.14) tends to EΔ(X1)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 Theorem2.1.
We assume thatEfi(Xi)=0. For large enough k, we have
(3.15)kε1<kε2ln(1+kε2).
In what follows, we suppose that kε1 is an integer for notation convenience. In the sequel, we take a specific sequence {εk}k≥1 as in Lemma 3.2, and then (3.15) amounts to kε1<(εkln(1+εk-1))-1.
We define a sequence{Yk,j}1≤j≤k-kε1 of random variables as follows:
(3.16)Yk,j=fkε1+j(Xkε1+j)I{|fkε1+j(Xkε1+j)|≤εksk}+εksk-εkskI{fkε1+j(Xkε1+j)<-εksk}+εkskI{fkε1+j(Xkε1+j)>εksk}.
Hence, we have 0≤Yk,j≤2εksk and {Yk,j≤x}={fkε1+j(Xkε1+j)≤-εksk+x} for 0≤x≤2εksk. The inequality (2.3) then implies that, for large enough k,
(3.17)sup{|P(⋂i=1j{Yk,vi≤xvi})-∏i=1j‍P(Yk,vi≤xvi)|:(xv1,…,xvj)∈[0,2εksk]j}≤(1-k-ε1)k-kε1-j.
For k≥1, denote ck2=∑j=1k-kε1‍Var(Yk,j). Involving Lemma 3.2, it follows that ck2/sk2→1 as k→∞.
Next, we calculate the difference between characteristic functions of sums of independent and dependent random variables. Takinggj(·)=exp(it(·/ck)) and M=2εksk in Lemma 3.1, where i=-1, we obtain
(3.18)|E(exp(itck∑j=1k-kε1‍(Yk,j-EYk,j)))-∏j=1k-kε1‍E(exp(itck(Yk,j-EYk,j)))|≤|∑j=2k-kε1‍∑1≤v1<⋯<vj≤k-kε1‍(-1)jexp(2εkskckit(k-kε1-j))·∫[0,2εksk]j‍(itck)jexp(itck(xv1+⋯+xvj))·(P(⋂i=1j{Yk,vi≤xvi})-∏i=1j‍P(Yk,vi≤xvi))dxv1⋯dxvj|≤c∑j=2k-kε1‍(k-kε1j)(1-k-ε1)k-kε1-j|2tk-ε2(skck)|j≤c(1-k-ε1+2|t|k-ε2skck)k-kε1,
where the second inequality comes from (3.17) and c is a positive constant. Since ε1<ε2 and sk/ck→1 as k→∞, (3.18) tends to 0 as k→∞.
Define a sequence{Wj}1≤j≤k-kε1 of independent random variables such that Wj has the same distribution with Yj. By our construction, for sufficiently large k,
(3.19){|Yk,j-EYk,j|=εkck}={|fkε1+j(Xkε1+j)-Efkε1+j(Xkε1+j)|=εksk}.
Involving Lemma 3.2, it yields that Yk,j satisfies the Lindeberg condition. Accordingly, Wj also satisfies the Lindeberg condition. Thus, the central limit theorem is true for Wj since (3.18) approaches 0, which implies that the central limit theorem holds for Yk,j, that is,
(3.20)1ck∑j=1k-kε1‍(Yk,j-EYk,j)→DN(0,1)
as k→∞. Taking into account that sk/ck→1 and the definition of Yk,j, by (3.18) we know that
(3.21)1sk∑j=kε1+1k‍fj(Xj)→DN(0,1).
On the other hand, it is obvious that
(3.22)1sk∑i=1kε1‍∫‍|fi(Xi)|dP=1sk∑i=1kε1‍∫{|fi(Xi)|≤εksk}‍|fi(Xi)|dP+1sk∑i=1kε1‍∫{|fi(Xi)|>εksk}‍|fi(Xi)|dP.
The first quantity on the right-hand side of (3.22) is up-bounded by
(3.23)1skεkskkε1=εkkε1≤1ln(1+εk-1)→0
as k→∞, using (3.15). Combining (3.5), (3.22), and (3.23), we obtain
(3.24)1sk∑i=1kε1‍fi(Xi)→L10
as k→∞. Therefore, (3.21) and (3.24) imply that
(3.25)1sk∑i=1k‍fi(Xi)→DN(0,1),
which concludes the proof of Theorem 2.1.Proof of Theorem2.3.
We defineYk,j, ck, and εk as in the proof of Theorem 2.1. We obtain
(3.26)|E(exp(itck∑j=1k-kε1‍(Yk,j-EYk,j)))-∏j=1k-kε1‍E(exp(itck(Yk,j-EYk,j)))|≤c∑j=2k-kε1‍(k-kε1j)(1-k-ε1)k-kε1-j|tck|j≤c(1-k-ε1+|tck|)k-kε1,
where c is a positive constant. Since sk=Θ(k) and sk/ck→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.



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