- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 294910, 5 pages
A New Proof of Central Limit Theorem for i.i.d. Random Variables
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.
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).
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  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 , Feller [2, 3], Adams , Billingsley , Dalang , Dudley , Nourdin and Peccati , Ho and Chen , 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).
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. .
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 .
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 . For a small but fixed , let be the unique viscosity solution of
By Lemma 5,
Since (21) is a uniformly parabolic PDE, thus by the interior regularity of (see Wang ), we have
We set and . Then
By Taylor’s expansion,
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
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.
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.
- J. Lindeberg, “Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung,” Mathematische Zeitschrift, vol. 15, no. 1, pp. 211–225, 1922.
- W. Feller, An Introduction to Probability Theory and Its Applications, vol. 2 of 2nd edition, John Wiley & Sons, New York, NY, USA, 1971.
- W. Feller, “The fundamental limit theorems in probability,” Bulletin of the American Mathematical Society, vol. 51, no. 11, pp. 800–832, 1945.
- W. J. Adams, The Life and Times of the Central Limit Theorem, vol. 35 of History of Mathematics, Kaedmon Publishing, New York, NY, USA, 2nd edition, 2009.
- P. Billingsley, Probability and Measure, John Wiley & Sons, New York, NY, USA, 3rd edition, 1995.
- R. C. Dalang, “Une démonstrationélémentaire du théorème central limite,” Elemente der Mathematik, vol. 60, no. 1, pp. 1–9, 2005.
- R. M. Dudley, Real Analysis and Probability, Cambridge University Press, New York, NY, USA, 2nd edition, 2002.
- I. Nourdin and G. Peccati, Normal Approximations with Malliavin Calculus, vol. 192 of From Stein's Method to Universality, Cambridge University Press, New York, NY, USA, 2012.
- S. T. Ho and L. H. Y. Chen, “An bound for the remainder in a combinatorial central limit theorem,” Annals of Probability, vol. 6, no. 2, pp. 231–249, 1978.
- S. G. Peng, “Law of large numbers and central limit theorem under nonlinear expectations,” http://arxiv.org/abs/math/0702358.
- S. G. Peng, “A new central limit theorem under sublinear expectations,” http://arxiv.org/abs/0803.2656.
- M. G. Crandall, H. Ishii, and P. L. Lions, “User's guide to viscosity solutions of second order partial differential equations,” Bulletin of the American Mathematical Society, vol. 27, no. 1, pp. 1–67, 1992.
- L. H. Wang, “On the regularity theory of fully nonlinear parabolic equations: II,” Communications on Pure and Applied Mathematics, vol. 45, no. 2, pp. 141–178, 1992.