Research Article A Note on the Central Limit Theorems for Dependent Random Variables

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1 International Scholarly Research Network ISRN Probability and Statistics Volume 22, Article ID 92427, pages doi:.542/22/92427 Research Article A Note on the Central Limit Theorems for Dependent Random Variables Yilun Shang Institute for Cyber Security, University of Texas at San Antonio, San Antonio, TX 78249, USA Correspondence should be addressed to Yilun Shang, shylmath@hotmail.com Received 25 June 22; Accepted 3 September 22 Academic Editors: D. Fiems, A. Hutt, and M. Montero Copyright q 22 Yilun Shang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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, which is critical to understand inferential statistics and hypothesis testing 2, 3. The assumption of independence for a sequence of observations X,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 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 4 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 2, 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

2 2 ISRN Probability and Statistics Theorem and Theorem 2 of 2 and can be used to describe systems which are globally determined but locally random. It is noteworthy that the work 2 has been extended in another direction, where the sum of a random number of random variables is examined 5. The rest of the paper is organized as follows. In Section 2, we present our central limit theorems, and in Section 3, wegivetheproofs. 2. 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,X 2,... and Z be random D variables. The image of Z is denoted by I Z R. We write X n Z meaning that Xn converges in distribution to Z as n. Denote by N, the standard normal variable. Let E Z or simply EZ and Var Z represent the mean and variance of Z, respectively. Theorem 2.. Let {X i } i be a sequence of identically distributed random variables and {f i } i a sequence of measurable functions such that f i : I X R satisfying, for all x I X,either α x γ β f i x α 2 x γ 2 β 2, 2. or α x γ β f i x α 2 x γ 2 β 2, 2.2 for some γ i > and α i,β i R, i, 2. Assume that α EX γ β, α 2 EX γ 2 β 2, and E X γ γ 2 2 ε < for some ε>. Let <ε <ε/ 2 ε and s 2 k k Var f i X i. Iffor sufficiently large k j sup{ P { } fvi X vi x vi k ε k k ε j, P } f vi X vi x vi : x v,...,x vj R j 2.3 where v,...,v j is any choice of indices such that k ε <v < <v j k, then s k fi X i Ef i X i D N, 2.4 as k. Corollary 2.2. Let {X i } i be a sequence of identically distributed random variables and {f i } i a sequence of measurable functions such that f i : I X R satisfying, for all x I X, expressions

3 ISRN Probability and Statistics 3 2. or 2.2. Assume that E X n < for n N. Suppose there exists δ, <δ<, such that for sufficiently large k the inequality j sup{ P { } fvi X vi x vi k δ/2 k k δ/2 j P } f vi X vi x vi : x v,...,x vj R j 2.5 holds, where v,...,v j is any choice of indices such that k δ/2 <v < <v j k,then s k fi X i Ef i X i D N, 2.6 as k. Generally, {f i } i is a sequence of nonlinearly bounded functions. If we set α α 2 γ γ 2 andβ β 2, then f i x xfor all i, which is the special case considered in 2 Theorem 3. It is worth noting that in the conditions of Theorem 2. we used the expression X γ, where γ is a real number. Naturally, we preclude the situation where the random variable X takes negative values and γ 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, then the right-hand side of 2.3 is close to. 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. 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 be a sequence of identically distributed random variables and {f i } i a sequence of measurable functions such that f i : I X R satisfying, for all x I X,either α x γ β f i x α 2 x γ 2 β 2, 2.7 or α x γ β f i x α 2 x γ 2 β 2, 2.8

4 4 ISRN Probability and Statistics for some γ i > and α i,β i R, i, 2. Assume that α EX γ β, α 2 EX γ 2 β 2, and E X γ γ 2 2 ε < for some ε>. Let <ε <ε/ 2 ε and s 2 k k Var f i X i. Iffor sufficiently large k R j j P { } fvi X vi x vi k ε k k ε j, P f vi X vi x vi dx v dx vj 2.9 where v,...,v j is any choice of indices such that k ε <v < <v j k, then s k fi X i Ef i X i D N, 2. as k. Corollary 2.4. Let {X i } i be a sequence of identically distributed random variables and {f i } i a sequence of measurable functions such that f i : I X R satisfying, for all x I X, expressions 2.7 or 2.8. Assume that E X n < for n N. Suppose there exists δ, <δ<, such that for sufficiently large k the inequality R j j P { } fvi X vi x vi k δ/2 k k δ/2 j P f vi X vi x vi dx v dx vj 2. holds, where v,...,v j is any choice of indices such that k δ/2 <v < <v j k,then s k fi X i Ef i X i D N, 2.2 as k. 3. Proofs We will prove Theorems 2. and 2.3 in this section through several lemmas, and the proof of corollaries follows directly. For convenience, denote by k a i / a v a vj, v < < v j k, the product of all a i s except a v,...,a vj. The following lemma regarding the difference between characteristic functions of sums of dependent and independent random variables is stated in 2 without proof.

5 ISRN Probability and Statistics 5 Lemma 3.. Let X,X 2,...,X k be random variables satisfying X i M, i k. Let g,...,g k :,M C be absolutely continuous integrable functions. Then one has the identity k E g i X i k Eg i X i j 2 v < <v j k k j g i M g v M g vj M j P {X vi x vi } M M g v i x vi P X vi x vi dx v dx vj. 3. Proof. Let F i x P X i x for i. Therefore, for k, we have E g X M g M g x df x M F x g x dx 3.2 by integration by parts. Let F x,x 2 P X x,x 2 x 2. Then, for k 2, we derive from 3.2 that E g X g 2 X 2 M Eg X Eg 2 X 2 g x g 2 x 2 F x,x 2 dx dx 2 M g x F x dx M M g x g 2 x 2 g 2 x 2 F 2 x 2 dx 2 F x,x 2 F x F 2 x 2 dx dx The general results then follow by induction. The following lemma compares with the Lindeberg condition. Lemma 3.2. Let {X i } i be a sequence of identically distributed random variables and f i : I X R satisfying, for all x I X, expressions 2. or 2.2, i. Assume that α EX γ β,

6 6 ISRN Probability and Statistics α 2 EX γ 2 β 2, and E X γ γ 2 2 ε < for some ε>. Let <ε <ε/ 2 ε. Then, there exists a sequence of reals {ε k } k such that s k s 2 k ε k, ε k s k as k, 3.4 { f i X i >ε k s k } { f i X i >ε k s k } f i X i dp as k, 3.5 f 2 i X i dp as k. 3.6 Proof. Without loss of generality, we can assume that f i X i are centered at, that is, Ef i X i. In fact, let f i : f i Ef i X, and then Ef i X i. By the assumptions 2. and 2.2, we have <α EX γ β Ef i X α 2 EX γ 2 β 2 <. 3.7 Therefore, f i : I X R satisfies α x γ β α 2 EX γ 2 β 2 f i x α 2x γ 2 β 2 α EX γ β. 3.8 For x I X,if 2. holds, then α x γ β α 2 EX γ 2 β 2 ; if 2.2 holds, then α 2 x γ 2 β 2 α EX γ β. Furthermore, by assumptions, we have α EX γ β α 2 EX γ 2 β 2 and α 2 EX γ 2 β 2 α EX γ β. Hence, we may assume that Ef i X i without loss of generality. Choose ε 2 so that ε <ε 2 <ε/ 2 ε. Define ε k k ε 2 for k 3. We will show that the constructed sequence {ε k } k is what we want. Since s 2 k k Var f i X i and X i s are identically distributed, by virtue of the bounds in 2. and 2.2 we obtain kc 2 s2 k kc2 2, 3.9 where C 2 E α2 X2γ 2α β X γ β2 α2 2 X2γ 2 2α 2 β 2 X γ 2 β2 2 and C2 2 E α2 X2γ 2α β X γ β 2 α2 2 X2γ 2 2α 2 β 2 X γ 2 β2 2. By our assumptions, we have k ε k s k C k ε 2 3.

7 ISRN Probability and Statistics 7 as k, which verifies 3.4. LetΔ X : α X γ β α 2 X γ 2 β 2. We have s k { f i X i >ε k s k } k s k {Δ X >ε k s k } fi X i dp Δ X dp k ε Δ X 2 ε dp. s k ε k s k {Δ X >ε k s k } 3. Since ε 2 <ε/ 2 ε and C k sk C 2 k, the quantity before the integration in 3. tends to zero. Consequently, 3.5 readily holds since the integration in 3. is bounded using our assumptions. Now we want to verify 3.6.Notethat f 2 i X i dp k Δ X { f i X i >ε k s k } s 2 2 dp k {Δ X >ε k s k } O Δ X 2 dp. {Δ X >ε k s k } s 2 k 3.2 By the Markovian inequality see e.g.,, P Δ X 2 >ε 2 k s2 k EΔ X 2 ε 2 k s2 k 3.3 as k. We have EΔ X 2 Δ X 2 dp Δ X 2 dp, {Δ X 2 ε 2 k s2 k } {Δ X 2 >ε 2 k s2 k } 3.4 where the first integration in 3.4 tends to EΔ X 2 < involving 3.3. Therefore, the second integration in 3.4 tends to zero, which, together with 3.2, finally verifies 3.6. Proof of Theorem 2.. We assume that Ef i X i. For large enough k, we have k ε < k ε 2 ln k ε In what follows, we suppose that k ε is an integer for notation convenience. In the sequel, we take a specific sequence {ε k } k as in Lemma 3.2, and then 3.5 amounts to k ε < ε k ln ε k.

8 8 ISRN Probability and Statistics We define a sequence {Y k,j } j k k ε of random variables as follows: Y k,j f k ε j Xk ε j I{ fk ε j X k ε j ε k s k } ε k s k ε k s k I {fk ε j X k ε j < ε k s k } ε k s k I {fk ε j X k ε j >ε k s k }. 3.6 Hence, we have Y k,j 2ε k s k and {Y k,j x} {f k ε j X k ε j ε k s k x} for x 2ε k s k. The inequality 2.3 then implies that, for large enough k, j sup{ P {Y k,vi x vi } k ε k k ε j. P Y k,vi } x vi : x v,...,x vj, 2ε k s k j 3.7 For k, denote c 2 k k k ε j Var Y k,j. Involving Lemma 3.2, it follows that c 2 k /s2 k as k. Next, we calculate the difference between characteristic functions of sums of independent and dependent random variables. Taking g j exp it / and M 2ε k s k in Lemma 3., where i, we obtain E exp it k k ε k k ε j j 2 v < <v j k k ε k k ε Yk,j EY k,j j it E exp Yk,j EY k,j j s k exp 2ε k it k k ε j it j it exp x v x vj,2ε k s k j j P {Y k,vi x vi } P Y k,vi x vi dx v dx vj k k ε k k ε c k ε k k ε j j 2tk ε 2 j 2 c k ε 2 t k ε s k k ε 2 k, sk j 3.8 where the second inequality comes from 3.7 and c is a positive constant. Since ε <ε 2 and s k / ask, 3.8 tendstoask. Define a sequence {W j } j k k ε of independent random variables such that W j has the same distribution with Y j. By our construction, for sufficiently large k, { Y k,j EY k,j ε k } { f k ε j Xk ε j Efk ε j Xk ε j ε k s k }. 3.9

9 ISRN Probability and Statistics 9 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 3.8 approaches, which implies that the central limit theorem holds for Y k,j,thatis, k k ε j Yk,j EY k,j D N, 3.2 as k. Taking into account that s k / that and the definition of Y k,j,by 3.8 we know s k j k ε f j Xj D N,. 3.2 On the other hand, it is obvious that k ε f i X i dp k ε s k s k k ε s k { f i X i ε k s k } { f i X i >ε k s k } f i X i dp fi X i dp The first quantity on the right-hand side of 3.22 is up-bounded by s k ε k s k k ε ε k k ε ln ε k 3.23 as k,using 3.5. Combining 3.5, 3.22,and 3.23, weobtain k ε f i X i L s k 3.24 as k. Therefore, 3.2 and 3.24 imply that s k f i X i D N,, 3.25 which concludes the proof of Theorem 2..

10 ISRN Probability and Statistics Proof of Theorem 2.3. We define Y k,j,,andε k as in the proof of Theorem 2.. Weobtain E exp it k k ε j k k ε Yk,j EY k,j j k k ε k k ε c k ε k kε j t j j 2 c k ε t k kε, it E exp Yk,j EY k,j j 3.26 where c is a positive constant. Since s k Θ k and s k / ask, the expression 3.26 tends to as k. The remaining part of the proof follows as in the proof of Theorem 2.. References P. Billingsley, Probability and Measure, John Wiley & Sons, New York, NY, USA, 3rd edition, A. DasGupta, Asymptotic Theory of Statistics and Probability, Springer, New York, NY, USA, T. S. Ferguson, A Course in Large Sample Theory, Chapman & Hall, New York, NY, USA, K. N. Berk, A central limit theorem for m-dependent random variables with unbounded m, Annals of Probability, vol., no. 2, pp , T. C. Christofides and P. M. Mavrikiou, Central limit theorem for dependent multidimensionally indexed random variables, Statistics & Probability Letters, vol. 63, no., pp , P. H. Diananda, The central limit theorem for m-dependent variables, vol. 5, pp , W. Hoeffding and H. Robbins, The central limit theorem for dependent random variables, Duke Mathematical Journal, vol. 5, pp , Y. Shang, A central limit theorem for randomly indexed m-dependent random variables, Filomat, vol. 26, no. 4, pp , R. Balan and I.-M. Zamfirescu, Strong approximation for mixing sequences with infinite variance, Electronic Communications in Probability, vol., pp. 23, 26. J. Beran, Statistical methods for data with long-range dependence with discussion, Statistical Science, vol. 7, no. 4, pp , 992. R. C. Bradley, Basic properties of strong mixing conditions. A survey and some open questions, Probability Surveys, vol. 2, pp. 7 44, M. Kaminski, Central limit theorem for certain classes of dependent random variables, Theory of Probability and its Applications, vol. 5, no. 2, pp , E. Ould-Saïd and A. Tatachak, Strong consistency rate for the kernel mode estimator under strong mixing hypothesis and left truncation, Communications in Statistics, vol. 38, no. 8, pp , M. Rosenblatt, A central limit theorem and a strong mixing condition, Proceedings of the National Academy of Sciences of the United States of America, vol. 42, no., pp , Y. Shang, Central limit theorem for the sum of a random number of dependent random variables, Asian Mathematics and Statistics, vol. 4, no. 3, pp , 2.

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