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

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).

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Academic Editor:Xinguang Zhang

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

Received 13 November 2013; Accepted 16 December 2013

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.

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 G -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 _{C b , Lip} ( ^{... n} ) denote the class of bounded functions f satisfying [figure omitted; refer to PDF] for some C > 0 depending on f ; let C ( ^{... n} ) denote the class of continuous functions f ; let ^{C b 2,3} ( [ 0 , ∞ ) × ^{... n} ) denote the class of bounded and 2 -time continuously differentiable functions with bounded derivatives of all orders less than or equal to 2 on [ 0 , ∞ ) and 3 -time continuously differentiable functions with bounded derivatives of all orders less than or equal to 3 on ^{... n} .

Let X be a random variable with distribution function V , so that, for any y ∈ ... , [figure omitted; refer to PDF]

If f is any function in _{C b , Lip} ( ... ) , the mathematical expectation of f ( X ) exists and [figure omitted; refer to PDF]

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 { _{X i} ^{} i = 1 ∞} is a sequence of i.i.d. random variables. Then

(i) for each f ∈ C ( ... ) , if E [ | f ( _{X 1} ) | ] < ∞ , then ∀ i , j ∈ ... , [figure omitted; refer to PDF]

(ii) ∀ i ∈ ... ; for each f ∈ C ( ^{... i + 1} ) , if E [ | f ( Y , _{X i + 1} ) | ] < ∞ , then [figure omitted; refer to PDF] where Y : = ( _{X 1} , ... , _{X i} ) .

Proposition 2.

Suppose X is a normally distributed random variable with E [ X ] = 0 and E [ ^{X 2} ] = ^{σ 2} > 0 , denoted by N ( 0 , ^{σ 2} ) . Then if Y = d X and Y is independent of X , we have, for each f ∈ _{C b , L i p} ( ... ) , [figure omitted; refer to PDF]

We will show that a normally distributed random variable X with E [ X ] = 0 and E [ ^{X 2} ] = ^{σ 2} > 0 is characterized by the following PDE defined on [ 0 , ∞ ) × ... : [figure omitted; refer to PDF] with Cauchy condition u ( 0 , x ) = f ( x ) . Equation (7) is called the heat equation.

Definition 3.

A real-valued continuous function u ∈ C ( [ 0 , ∞ ) × ... ) is called a viscosity subsolution (resp., supersolution) for (7), if for each function v ∈ ^{C b 2,3} ( [ 0 , ∞ ) × ... ) and for each minimum (resp., maximum) point ( t , x ) ∈ [ 0 , ∞ ) × ... of v - u , we have [figure omitted; refer to PDF] u 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.

Let X be an N ( 0 , ^{σ 2} ) distributed random variable. For each f ∈ _{C b , L i p} ( ... ) , we define a function [figure omitted; refer to PDF]

Then we have [figure omitted; refer to PDF]

We also have the estimates: for each T > 0 , there exists a constant C > 0 such that, for all t , s ∈ [ 0 , T ] and x , y ∈ ... , [figure omitted; refer to PDF]

Moreover, u is the unique viscosity solution, continuous in the sense of (11) and (12), of (7) with Cauchy condition u ( 0 , x ) = f ( x ) .

Proof.

Since [figure omitted; refer to PDF] we then have (11). Let Y be independent of X such that Y = d X . By Propositions 1 and 2, we have [figure omitted; refer to PDF]

It follows from this and (11) that [figure omitted; refer to PDF] which implies (12).

Now, for a fixed point ( t , x ) ∈ ( 0 , ∞ ) × ... , let v ∈ ^{C b 2,3} ( [ 0 , ∞ ) × ... ) satisfy v ...5; u and v ( t , x ) = u ( t , x ) . By (10), we have, for δ ∈ ( 0 , t ) , [figure omitted; refer to PDF] where C ¯ is a positive constant, and then, we have [figure omitted; refer to PDF]

Hence, u is a viscosity subsolution for (7). Similarly, we can prove that u 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 { _{X i} ^{} i = 1 ∞} be a sequence of i.i.d. random variables. We further assume that [figure omitted; refer to PDF]

Denote _{S n} ... = ^{∑ i = 1 n} ( _{X i} - μ ) . Then [figure omitted; refer to PDF]

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

Lemma 7.

Under the assumptions of Theorem 6, we have [figure omitted; refer to PDF] for any f ∈ _{C b , L i p} ( ... ) , where X is N ( 0 , ^{σ 2} ) .

Proof.

The main approach of the following proof derives from Peng [10]. For a small but fixed h > 0 , let v be the unique viscosity solution of [figure omitted; refer to PDF]

By Lemma 5, [figure omitted; refer to PDF]

Particularly, [figure omitted; refer to PDF]

Since (21) is a uniformly parabolic PDE, thus by the interior regularity of v (see Wang [13]), we have [figure omitted; refer to PDF]

We set δ ... = 1 / n and _{S 0} ... = 0 . Then [figure omitted; refer to PDF]

By Taylor's expansion, [figure omitted; refer to PDF]

Thus [figure omitted; refer to PDF]

We now prove that [figure omitted; refer to PDF]

Indeed, for the 3rd term of ^{J δ i} , by Proposition 1, [figure omitted; refer to PDF]

For the second term of ^{J δ i} , by Proposition 1, we have [figure omitted; refer to PDF]

Thus combining the above two equalities with [figure omitted; refer to PDF] we have [figure omitted; refer to PDF]

Thus, (27) can be rewritten as [figure omitted; refer to PDF]

But since both _{∂ t} v and ^{∂ x x 2} v are uniformly α -hölder continuous in x and α / 2 -hölder continuous in t on [ 0,1 ] × ... , we then have [figure omitted; refer to PDF]

Thus [figure omitted; refer to PDF] where C is a positive constant. As n [arrow right] ∞ , we have [figure omitted; refer to PDF]

On the other hand, for each t , ^{t [variant prime]} ∈ [ 0,1 + h ] and x ∈ ... , [figure omitted; refer to PDF]

Thus [figure omitted; refer to PDF] and by (23) [figure omitted; refer to PDF]

It follows from (23), (36), (38), and (39) that [figure omitted; refer to PDF]

Since h can be arbitrarily small, we have [figure omitted; refer to PDF]

Proof of Theorem 6.

For notional simplification, write [figure omitted; refer to PDF]

Let [straight epsilon] be any positive number, and take δ small enough such that V ¯ ( y + δ ) - V ¯ ( y - δ ) ...4; [straight epsilon] . Construct two functions f , g such that [figure omitted; refer to PDF]

Then [figure omitted; refer to PDF] and for each n , [figure omitted; refer to PDF]

Obviously, f and g ∈ _{C b , Lip} ( ... ) . By Lemma 7, we have [figure omitted; refer to PDF]

So [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF]

Since this is true for every [straight epsilon] , we have [figure omitted; refer to PDF]

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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