14CHAPTER 8.BOOTSTRAP AND JACKKNIFE ESTIMATION OF SAMPLING DISTRIBUTIONS Theorem 4.2 (Singh,1981). A.If E(X2)<oo,then Dm=Dn(X)=lP*(v元(n-xn)≤x)-P(元（区n-EF(X)≤x)llo→as.0. B.If E(X)<oo,then P (loglognDVar √n lim sup a.s. 2σ2V2re C.If EX3<oo,and D%=lP(Vm(,n-xn)/Sn≤x)-P(V(区n-Er(X)/o≤x)ll where S号=n-1∑r(Xi-Xn)2,then limsupv元DR≤Kp/a3 a.s. where p=EX-u3<oo and K is the universal constant of the Berry-Esseen bound. D.If EX3<oo and F is non-lattice,then P"vC-X/.≤=+n是+on% uniformly in x a.s.where and o are the standard normal d.f.and standard normal density function respectively;hence in this case VnDia.s.0. Now we turn to the corresponding behavior of the bootstrap empirical distribution function F(or bootstrap empirical measure P).We know that for =R we have,by the inverse transformation, V元(Fn-F)Un(F)→U(F) where Un is the empirical process of n i.i.d.Uniform(0,1)random variables and U is a Brownian bridge process on [0,1].The following theorem says that the bootstrap mimics this behavior for almost every sequence X1,X2,.... Theorem 4.3 If m An->oo,then for almost every sequence X1,X2,..., vm(Fn-Fn)→U*(F) where U*is a Brownian bridge process on [0,1]. Proof.The following proof is due to Shorack (1982).Let 6i,,..be i.i.d.Uniform(0,1),let Gm be the empirical d.f.of the first m of the 's,and let Um=vm(G-I)be the corresponding empirical process.By the Skorokhod construction we can construct the sequence {U}on a common probability space with a Brownian bridge process U*so that llUm-U*lloo-a.s.0.[In fact by the Hungarian construction,this can be carried out with a sequence of Brownian bridge processes Bo so thatmlM(logm)/m almost surely;at the moment we only need the less precise result.14CHAPTER 8. BOOTSTRAP AND JACKKNIFE ESTIMATION OF SAMPLING DISTRIBUTIONS Theorem 4.2 (Singh, 1981). A. If E(X2 ) < ∞, then Dn ≡ Dn(X) ≡ kP ∗ ( √ n(X ∗ n − Xn) ≤ x) − P( √ n(Xn − EF (X)) ≤ x)k∞ →a.s. 0. B. If E(X4 ) < ∞, then lim sup n→∞ √ n (log log n) 1/2 Dn = p V ar[(X − µ) 2] 2σ 2 √ 2πe a.s. C. If E|X| 3 < ∞, and Ds n ≡ kP ∗ ( √ n(X ∗ n − Xn)/Sn ≤ x) − P( √ n(Xn − EF (X))/σ ≤ x)k∞ where S 2 n = n −1 Pn 1 (Xi − Xn) 2 , then lim sup n→∞ √ nDs n ≤ Kρ/σ3 a.s. where ρ ≡ E|X − µ| 3 < ∞ and K is the universal constant of the Berry - Esseen bound. D. If E|X| 3 < ∞ and F is non-lattice, then P ∗ ( √ n(X ∗ n − Xn)/Sn ≤ x) = Φ(x) + µ3(1 − x 2 ) 6σ 3n1/2 φ(x) + o(n −1/2 ) uniformly in x a.s. where Φ and φ are the standard normal d.f. and standard normal density function respectively; hence in this case √ nDs n →a.s. 0. Now we turn to the corresponding behavior of the bootstrap empirical distribution function F ∗ n (or bootstrap empirical measure P ∗ n ). We know that for X = R we have, by the inverse transformation, √ n(Fn − F) d= Un(F) ⇒ U(F) where Un is the empirical process of n i.i.d. Uniform(0, 1) random variables and U is a Brownian bridge process on [0, 1]. The following theorem says that the bootstrap mimics this behavior for almost every sequence X1, X2, . . .. Theorem 4.3 If m ∧ n → ∞, then for almost every sequence X1, X2, . . ., √ m(F ∗ m − Fn) ⇒ U ∗ (F) where U ∗ is a Brownian bridge process on [0, 1]. Proof. The following proof is due to Shorack (1982). Let ξ ∗ 1 , ξ∗ 2 , . . . be i.i.d. Uniform(0, 1), let G∗ m be the empirical d.f. of the first m of the ξ ∗ i ’s, and let U ∗ m ≡ √ m(G∗ m −I) be the corresponding empirical process. By the Skorokhod construction we can construct the sequence {U ∗ m} on a common probability space with a Brownian bridge process U ∗ so that kU ∗ m − U ∗k∞ →a.s. 0. [In fact by the Hungarian construction, this can be carried out with a sequence of Brownian bridge processes B 0 m so that kU ∗ m − B 0 mk∞ ≤ M(log m)/ √ m almost surely; at the moment we only need the less precise result.]