1 = δ2 (0, ), Y Y n nδ. , T n = Y Y n n. ( U n,k + X ) ( f U n,k + Y ) n 2n f U n,k + θ Y ) 2 E X1 2 X1

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1 8. The cetral limit theorems 8.1. The cetral limit theorem for i.i.d. sequeces. ecall that C ( is N -separatig. Theorem 8.1. Let X 1, X,... be i.i.d. radom variables with EX 1 = ad EX 1 = σ (,. Suppose that there is a radom variable X such that X X σ X i distributio. The, for ay i.i.d. radom variables Y 1, Y,... with EY 1 = ad EY 1 = δ (,, Y Y δ X i distributio. Proof. It loseo geerality to assume that σ = δ = 1 ad that (X =1 ad (Y =1 are idepedet. Set S = X X, T = Y Y. Clearly, E(S = E(T = 1. By Exercise 7.3, (S =1 ad (T =1 are mass-preservig. Note that Ef(S Ef(X for ay bouded cotiuous fuctio f. Sice C ( is N -separatig, to prove this theorem, it remais to show that Ef(S Ef(T for all f C (. Let f C (. For, set U, = X X 1 + Y Y ad V, = f ( U, + X ( f U, + Y. Clearly, f(s f(t = V,. By Taylor s theorem, there are θ, θ [, 1 such that (8.1 V, = X ( Y f (U, + X f U, + θ X Y ( f U, + θ Y. Set f (h = max{ f (x f (y : x y h. Sice f is uiformly cotiuous, f (h as h. Note that f (h f (h + δ f (h + f (δ. This implies that f is cotiuous ad, thus, Borel measurable. By (8.1, we obtai EV, 1 E [ (X Y f (U, 1 [ ( ( E X X f + Y Y f or, equivaletly, EV, 1 [ ( ( E X1 X1 f + Y Y1 1 f. Cosequetly, this yields Ef(S Ef(T 1 [ ( ( E X1 X1 f + Y Y1 1 f. By the Lebesgue domiated covergece theorem, Ef(S Ef(T a. The ext theorem is a simple corollary of Theorems 1.4 ad 8.1. Theorem 8. (The cetral limit theorem. Let X 1, X,... be i.i.d. radom variables with mea µ ad variace σ >. The, X X µ σ N(, 1 i distributio. 47

2 emar 8.1. Let X 1, X,... be i.i.d. Beroulli radom variables with P(X 1 = 1 = P(X 1 = 1 = 1/ ad X be a stadard ormal radom variable. Let f, f be the characteristic fuctios of X, X. Note that f (u = eiu + e iu = cos u, = 1,,... By the cetral limit theorem, we have ( u ( ( u f = cos f(u u. Usig Taylor s theorem, oe has, for fixed u, cos(u/ = (1 u (1 + O(, a. This leads to f(u = lim (1 u (1 + O( = e u /. 8.. The cetral limit theorem for o-idetical distributios. I this sectio, we itroduce two famous results related to the cetral limit theorem give by J. Lideburg ad W. Feller respectively i 19 ad For 1, let r be a positive itegers ad X,1,..., X,r be idepedet radom variables with mea ad variace Var(X, = σ,. Set S = r X,, s = Var(S = r σ,. The triagular array {X, : 1 r, 1 is said to possess (1 the Lideberg coditio (LC if, for all ϵ >, r 1 X, dp a. s { X, ϵ ( the uiform asymptotic egligibility (UAN if σ, max 1 r s a. (3 the cetral limit theorem (CLT if S D N(, 1, a. Theorem 8.3 (Lideberg(19. LC UAN + CLT. Theorem 8.4 (Feller(1935. UAN + CLT LC. emar 8.. Uder UAN, CLT LC. To prove the above theorems, we eed the followig settig. For 1 ad ϵ >, set A (ϵ = 1 r s X, dp (LC { X, ϵ B (ϵ = 1 r s 3 X, 3 dp { X, <ϵ 48

3 ad C = 1 s 4 r σ, 4, D σ, = max 1 r s (UAN. For 1 ad 1 r, let f, be the characteristic fuctio of X, ad defie r ( Σ (u = u f, e u σ, /(s Lemma 8.5. I the above settig, we have (1 B (ϵ ϵ. ( C D. (3 D A (ϵ + ϵ. (4 Σ (u u A (ϵ + u 3 B (ϵ + u 4 C. Proof. (1 ad ( are obvious. (3 follows immediately from the followig iequality. σ, s = 1 s X, dp + 1 { X, ϵ s X, dp A (ϵ + ϵ. { X, <ϵ To see (4, we eed the fact that, for ay radom variable Y with mea ad variace b >, Ee iy e b/ E (Y + b, where (y = e iy (1 + iy 1 y. Observe that e a 1 + a [, a / for a ad (Y (e iy e b/ = e b/ 1 iy + Y. eplacig a with b/ ad taig the expectatio o both sides gives the desired iequality. Next, let Y = ux,. This implies ( f, (u/ e u σ, /(s E ux, + u4 σ, 4. We will use the iequality (y y y 3 to boud the first term of the right side. To prove this iequality, it suffices to cosider y >, sice ( y = (y. Note that This implies (y = ie iy i + y, (y = e iy + 1, (y = ie iy. ( = ( = ( =, (y, (y = 1. Sice,, are cotiuously differetiable, we have that, for y, (y (y (y Cosequetly, we obtai ( E ux, ad this proves (4. y y y s 4 (z dz = y (y y (z dz (y (y / (z dz y (y 3 /6 y y 3 { X, ϵ ( ux, 49 dp + { X, <ϵ ux, 3 dp

4 Proof of Theorem 8.3. First, it is clear that LC is exactly the case A (ϵ for all ϵ > ad UAN is equivalet to D. By Lemma 8.5(3, LC implies UAN ad the Σ (u, a, for all u. To show the CLT, it is equivalet to prove r f, ( u e u /, where f, is the c.f. of X,. Let z 1,..., z ad w 1,..., w be complex umbers with absolute values at most 1. Note that z w z w. Lettig z = f, (u/ ad w = exp{ σ, u /(s implies r ( u f, e u / Σ (u, a. Defiitio 8.1. Ay triagular array {X, : 1 r, 1 with EX, = is said to have Lyapuov s coditio if L (δ = 1 s +δ r E X, +δ for some δ >. emar 8.3. Note that, for ay radom variable X, X dp X X δ ϵs dp E X +δ ϵ δ s δ. This implies X >ϵs X >ϵs Lyapuov s coditio LC. Corollary 8.6. Lyapuov s coditio implies CLT. Proof of Theorem 8.4. For 1 ad 1 r, let φ, (u = f, (u 1 ad r ( r [ u ψ (u = φ, + u = E e iux,/ 1 iux, + u σ, s. Step 1: A (ϵ ϵ 6 ψ ( 4 ϵ. To see this, observe that, for 1 r, ( [ [φ, + us u σ, = E e iux,/ 1 + u X, =E [ cos ( ux, 1 s X, X, ϵ s 1 + u X, ( u s ϵ dp. X, ϵ [ cos I the above computatios, the first iequality uses the fact s ( ux, t cos t 1 + t = (s si sds, t, u X, s dp

5 ad the secod iequality applies cos s 1. Summig up yields ( u ψ (u ϵ A (ϵ. The desired iequality is give by choosig u = 4/ϵ. Step : UAN max φ, (u/ for all u. First, observe that, for t >, (8. e it 1 = t ie is ds. This implies e it 1 t. Usig this fact, we have ( [ u φ, E e iux,/ 1 E P( X, ϵ + X, <ϵ ux, ux, dp σ, ϵ s + ϵ u Thus, for all ϵ >, max φ, (u/ ϵ D + ϵ u, 1 r which proves the desired property. Step 3: UAN r e φ,(u/ r f, (u/. Sice f, (v 1, we have φ, (v = f, (v 1 ad the e φ,(v = exp{φ, (v 1. Fix u. By Step, there is N > such that φ, (u/ < ϵ for 1 r ad N. This implies r r r e φ,(u/ f, (u/ e φ,(u/ f, (u/ = r e φ,(u/ r 1 φ, (u/ e ϵ φ, (u/, N, where the last iequality comes from the followig fact e α α 1 α = α! α e α, α C. = Moreover, by (8., oe has, for t, e it t 1 it = i(e is t 1ds e is 1 ds This implies r φ, (u/ ϵ r ϵu s r φ, (u/ ϵ r EX, = ϵu, N. This part is proved by the result i Step. Fially, if UAN ad CLT hold, the by Step 3, { r exp φ, (u/ e u / 51 t s drds = t. E eiux,/ 1 iux,

6 or equivaletly e ψ (u 1. This implies exp{ψ (u = e ψ (u 1, which gives ψ (u. By Step 1, LC follows D-Covergece uder UAN. Cosider the triagular array {X, 1 r, 1, where X,1,..., X,r are idepedet ad r EX, =, EX, = σ,, σ, = 1. Let f, be the characteristic fuctio of X, ad set S = r X,. D Assumptio: S L uder UAN. Note that { r r D S L fs = f, f L exp φ, f L where φ, = f, 1 ad the secod equivalece comes from step 3 i the proof of Theorem 8.4. Observe that, r (8.3 φ, (u = (e iux 1 iuxdf (x where F = r F, ad F, is the distributio fuctio of X,. For 1, set dν (x = x df (x. Clearly, ν ({ = ad ν is absolutely cotiuous w.r.t. F with ado derivative x. Note that F iot a probability, but ν is, because r r ν( = x df (x = x df, (x = σ, = 1. For u, set (8.4 h(u, x = By (8.3, we have r { (e iux 1 iux/x if x u / if x = φ, (u = h(u, xdν (x ecall that the class M is sequetially compact uder the D-covergece. Oe may choose ν M ad a subsequece (ν which D-coverges to ν. Clearly, h is cotiuous ad vaishes at ±. This implies h(u, xdν (x h(u, xdν(x. Exercise 8.1. Let µ N. Prove that µ D-coverges to µ M if ad oly if fdµ fdµ for all cotiuous fuctios vaishig at ±. Next, let σ = ν({ ad Q be a measure satisfyig dq(x = 1 dν(x for x ad x Q({ =. I this settig, we may rewrite h(u, xdν(x = σ u + (e iux 1 iuxdq(x. 5

7 This implies, for u, (8.5 f L (u = exp where Q({ = ad σ + x dq(x 1. { σ u + (e iux 1 iuxdq(x, Theorem 8.7. For 1, let X,1,..., X,r be idepedet radom variables with fiite variaces ad set S = r X,. Assume that UAN holds ad If S D L, the fl (u = e ψ(u with ES a, Var(S b <. ψ(u = iau σ b u where σ is a costat, K({ = ad x dk(x <. + (e iux 1 iuxdk(x, Proof. For 1 ad 1 r, set X, = X, EX,, S = Var(S r X, = S ES Var(S. Clearly, E X, = ad r E X, = 1. Sice S D L, ES a ad Var(S b, we have D L a S L =. b By (8.5, there is a costat σ > ad a measure Q o satisfyig Q({ = ad σ + x dq(x 1 such that f L(u = exp { σ u + (e iux 1 iuxdq(x. By settig dk(y = dq(y/b, we obtai f L (u = e iau f L(bu = exp {iau σ b u + emar 8.4. Cosider some particular cases. (1 If K = ad σ =, the L = a a.s.. (e iuy 1 iuydk(y. ( If K =, the L D = N(a, σ. (3 If a = σ = ad K = cδ x, the f L (u = exp { c(e iux 1 iux. Let Z λ be a Poisso radom variable with parameter λ >, i.e. P(Z λ = = e λ λ /! for =, 1,,... The f Zλ (u = exp { (e iu 1λ ad L D = x(z c c. (4 If a = σ = ad K = c δ x, the L = D x (Z c c, where Z c1,..., Z c are idepedet. Propositio 8.8. Let ψ(u = iau σ u / + (eiux 1 iuxdk(x, where K({ = ad x dk(x <. The, e ψ(u is a characteristic fuctio of some radom variable with fiite secod momet. 53

8 Proof. It loseo geerality to assume that a = σ =. Set dµ(x = x dk(x ad let h be the fuctio i (8.4. Clearly, µ( < ad (8.6 (e iux 1 iuxdk(x = h(u, xdµ(x. Sice h is uiformly cotiuous ad bouded o A for ay bouded set A, the right side of (8.6, as a fuctio of u, is cotiuous o. For 1, let x, = + for all =, 1,..., 1 ad set µ = 1 = Immediately, we have µ([x,, x, + 1 δ x,, h (u, x = h(u, xdµ (x = = 1 = 1 = h(u, x, µ([x,, x, + 1 h (u, xdµ(x h(u, x, 1 [x,,x, + 1 (x. h(u, xdµ(x, where the covergece is give by the Lebesgue domiated covergece theorem. (I fact, µ D-coverges to µ ad, for u, x h(u, x vaishes at ifiity. By Exercise 8.1, oe has the above covergece. Next, set c, = x, Q([x,, x, + 1 if, c, = ad K = 1 = c, δ x,, X = 1 = x, (Z c, c, where (Z λ λ> are idepedet Poisso radom variables with EZ λ = λ ad Z. The, h(u, xdµ (x = (e iux 1 iuxdk (x u µ([, 1 ad { exp (e iux 1 iuxdk (x = f X (u = E[e iux. Lettig implies { f X (u exp (e iux 1 iuxdk(x, u. By the cotiuity theorem, there is a radom variable X such that X X i distributio ad { f X (u = exp (e iux 1 iuxdk(x. Observe that EX By Corollary 7.6, oe has 1 = x, E(Z c, c, = µ([, µ([, 1/. EX = lim if EX = µ( <. 54