Abstract and Applied Analysis

Volume 2013 (2013), Article ID 294910, 5 pages

http://dx.doi.org/10.1155/2013/294910

Research Article

A New Proof of Central Limit Theorem for i.i.d. Random Variables

Zhaojun Zong and Feng Hu

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Received 13 November 2013; Accepted 16 December 2013

Academic Editor: Xinguang Zhang

Copyright © 2013 Zhaojun Zong and Feng Hu. 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.

Abstract.

Central limit theorem (CLT) has long and widely been known as a fundamental result in probability theory. In this note, we give a new proof of CLT for independent identically distributed (i.i.d.) random variables. Our main tool is the viscosity solution theory of partial differential equation (PDE).

1. Introduction

Central limit theorem (CLT) has long and widely been known as a fundamental result in probability theory. The most familiar method to prove CLT is to use characteristic functions. To a mathematician having been already familiar with Fourier analysis, the characteristic function is a natural tool, but to a student of probability or statistics, confronting a proof of CLT for the first time, it may appear as an ingenious but artificial device. Thus, although knowledge of characteristic functions remains indispensable for the study of general limit theorems, there may be some interest in an alternative way of attacking the basic normal approximation theorem. Indeed, due to the importance of CLT, there exist the numerous proofs of CLT such as Stein’s method and Lindeberg’s method. Let us mention the contribution of Lindeberg [1] which used Taylor expansions and careful estimates to prove CLT. For more details of the history of CLT and its proofs, we can see Lindeberg [1], Feller [2, 3], Adams [4], Billingsley [5], Dalang [6], Dudley [7], Nourdin and Peccati [8], Ho and Chen [9], and so on.

Recently, motivated by model uncertainties in statistics, finance, and economics, Peng [10, 11] initiated the notion of independent identically distributed random variables and the definition of -normal distribution. He further obtained a new CLT under sublinear expectations.

In this note, inspired by the proof of Peng’s CLT, we give a new proof of the classical CLT for independent identically distributed (i.i.d.) random variables. Our proof is short and simple since we borrow the viscosity solution theory of partial differential equation (PDE).

2. Preliminaries

In this section, we introduce some basic notations, notions, and propositions that are useful in this paper.

Let denote the class of bounded functions satisfying

for some depending on ; let denote the class of continuous functions ; let denote the class of bounded and-time continuously differentiable functions with bounded derivatives of all orders less than or equal to on and-time continuously differentiable functions with bounded derivatives of all orders less than or equal to on .

Let be a random variable with distribution function , so that, for any ,

If is any function in , the mathematical expectation of exists and

Our proof is based on the following classical results for i.i.d. random variables and normally distributed random variables with zero means.

Proposition 1. Suppose is a sequence of i.i.d. random variables. Then

(i)for each , if , then ,

(ii); for each , if , then

where .

Proposition 2. Suppose is a normally distributed random variable with and , denoted by . Then if and is independent of , we have, for each ,

We will show that a normally distributed random variable with and is characterized by the following PDE defined on :

with Cauchy condition . Equation (7) is called the heat equation.

Definition 3. A real-valued continuous function is called a viscosity subsolution (resp., supersolution) for (7), if for each function and for each minimum (resp., maximum) point of , we have

is called a viscosity solution for (7) if it is both a viscosity subsolution and a viscosity supersolution.

Remark 4. For more basic definitions, results, and related literature on viscosity solutions of PDEs, the readers can refer to Crandall et al. [12].

Lemma 5. Letbe an distributed random variable. For each , we define a function

Then we have

We also have the estimates: for each , there exists a constant such that, for all and , ,

Moreover, is the unique viscosity solution, continuous in the sense of (11) and (12), of (7) with Cauchy condition .

Proof. Since

we then have (11). Letbe independent of such that . By Propositions 1 and 2, we have

It follows from this and (11) that

which implies (12).

Now, for a fixed point , let satisfy and . By (10), we have, for ,

where is a positive constant, and then, we have

Hence, is a viscosity subsolution for (7). Similarly, we can prove that is a viscosity supersolution for (7). The proof of Lemma 5 is completed.

3. A New Proof of CLT for i.i.d. Random Variables

Theorem 6. Let be a sequence of i.i.d. random variables. We further assume that

Denote . Then

In order to prove Theorem 6, we need the following lemma.

Lemma 7. Under the assumptions of Theorem 6, we have

for any , where is .

Proof. The main approach of the following proof derives from Peng [10]. For a small but fixed , let be the unique viscosity solution of

By Lemma 5,

Particularly,

Since (21) is a uniformly parabolic PDE, thus by the interior regularity of (see Wang [13]), we have

We set and . Then

By Taylor’s expansion,

Thus

We now prove that

Indeed, for the 3rd term of , by Proposition 1,

For the second term of , by Proposition 1, we have

Thus combining the above two equalities with

we have

Thus, (27) can be rewritten as

But since both and are uniformly -hölder continuous in and -hölder continuous in on , we then have

Thus

where is a positive constant. As , we have

On the other hand, for each , and ,

Thus

and by (23)

It follows from (23), (36), (38), and (39) that

Since can be arbitrarily small, we have

Proof of Theorem 6. For notional simplification, write

Let be any positive number, and take small enough such that . Construct two functions , such that

Then

and for each ,

Obviously, and . By Lemma 7, we have

So

Hence

Since this is true for every , we have

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their careful reading of this paper, correction of errors, and valuable suggestions. The authors thank the partial support from the National Natural Science Foundation of China (Grant nos. 11301295 and 11171179), the Doctoral Program Foundation of Ministry of Education of China (Grant nos. 20123705120005 and 20133705110002), the Postdoctoral Science Foundation of China (Grant no. 2012M521301), the Natural Science Foundation of Shandong Province of China (Grant nos. ZR2012AQ009 and ZR2013AQ021), and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province of China.

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