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

and Applied Analysis 3 3. A New Proof of CLT for i.i.d. Random Variables Theorem 6. Let {Xi} ∞ i=1 be a sequence of i.i.d. random variables. We further assume that E [X1] = μ, E [(X1 − μ) 2 ] = σ 2 > 0, E [ 󵄨󵄨󵄨󵄨X1 󵄨󵄨󵄨󵄨 3 ] < ∞. (18) Denote Sn := ∑ n i=1 (Xi − μ). Then lim n→∞ Pr( Sn √n ≤ y) = ∫ y −∞ 1 √2πσ e −x 2 /2σ 2 dx. (19) In order to proveTheorem 6, we need the following lemma. Lemma 7. Under the assumptions of Theorem 6, we have lim n→∞ E[f( Sn √n )] = E [f (X)] , (20) for any f ∈ Cb,Lip(R), whereX isN(0, σ). Proof. Themain approach of the following proof derives from Peng [10]. For a small but fixed h > 0, let V be the unique viscosity solution of ∂tV + 1 2 σ 2 ∂ 2 xx V = 0, (t, x) ∈ [0, 1 + h] ×R, V (1 + h, x) = f (x) . (21) By Lemma 5, V (t, x) = E [f (x + √1 + h − tX)] . (22) Particularly, V (h, 0) = E [f (X)] , V (1 + h, x) = f (x) . (23) Since (21) is a uniformly parabolic PDE, thus by the interior regularity of V (see Wang [13]), we have ‖V‖C1+α/2,2+α([0,1]×R) < ∞, for some α ∈ (0, 1) . (24) We set δ := 1/n and S0 := 0. Then V (1, δSn) − V (0, 0) = n−1 ∑ i=0 {V ((i + 1) δ, δSi+1) − V (iδ, δSi)} = n−1 ∑ i=0 {[V ((i + 1) δ, δSi+1) − V (iδ, δSi+1)] + [V (iδ, δSi+1) − V (iδ, δSi)]} = n−1 ∑ i=0 {I i δ + J i δ } . (25) By Taylor’s expansion, J i δ = ∂tV (iδ, δSi) δ + 1 2 ∂ 2 xx V (iδ, δSi) (Xi+1 − μ) 2 δ + ∂xV (iδ, δSi) (Xi+1 − μ)√δ,


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).

Preliminaries
In this section, we introduce some basic notations, notions, and propositions that are useful in this paper.
Let  , Lip (R  ) denote the class of bounded functions  satisfying for some  > 0 depending on ; let (R  ) denote the class of continuous functions ; let  2,3  ([0, ∞) × R  ) 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 R  .
Let  be a random variable with distribution function , so that, for any  ∈ R, Pr ( ≤ ) =  () . ( If  is any function in  , Lip (R), 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.
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].

A New Proof of CLT for i.i.d. Random Variables
Theorem 6.Let {  } ∞ =1 be a sequence of i.i.d.random variables.We further assume that In order to prove Theorem 6, we need the following lemma.