Oo the convergence to types theorem 871 discussed

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oo, the convergence to types Theorem 8.7.1 discussed below assures us we can replace the square root in the expression for by .Jk;n and thus we consider the density .Jk fn(.Jk n n n By Stirling's formula (see Exercise 8 of Chapter 9), as n _... oo, n! Jn <k _ 1 >,<n _ k)! _ ·
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8.2 Scheffe's lemma 257 Neglecting the factorials in the expression for the density, we have two factors of the form Thus we get for the density of the following asymptotic expression _1_(1 + _:_l-1(1- X )n-k. $ -/k (n -k)j-/k It suffices to prove that or equivalently, X X x 2 (k- 1) log(1 + r,:) + (n - k) log(1- -/k) - 2 . (8.7) vk (n- k)/ k Observe that, for It I < 1, and therefore oo tn -log(1- t) = L -;• n=l r2 o(t) :=1 - log(1- t)- <r + 2 >I 00 ltl 3 <"" itln = -- < 21t1 3 - L... 1-ltl- ' n=3 if It I < 1/2. So the left side of (8.7) is of the form where X X o(1) = (k- 1)8( rr:) + (n - k)o( rr:) o. vk (n- k)/vk Neglecting o(1), (8.7) simplifies to x x 2 1 1 2 --- -(1- -+ -n-) -x /2. -lk 2 k k-1 (8.8) 0
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258 8. Convergence in Distribution 8.3 The Baby Skorohod Theorem Skorohod's theorem is a conceptual aid which makes certain weak convergence results easy to prove by continuity arguments. The theorem is true in great gener- ality. We only consider the result for real valued random variables and hence the name Baby Skorohod Theorem . We begin with a brief discussion of the relationship of almost sure convergence and weak convergence. Proposition 8.3.1 Suppose {X, Xn, n 2:: 1} are random variables. If X a. s.x n-+ ' then Xn =>X. Proof. Suppose X n X and let F be the distribution function of X. Set so that P(N) = 0. For any h > 0 and x e C(F), we have the following set containments: c limsup[Xn ::: x] n Nc n-+oo and hence, taking probabilities F(x- h):::; P(lim inf[Xn ::: x]) n-+00 ::: liminfP[Xn::: x] n-+00 (from Fatou 's lemma) :::;limsupP[Xn :::x] n-+oo :::; P(lim sup[Xn ::: x]) (from Fatou 's lemma) n-+oo ::: F(x). Since x e C(F), let h .J.. 0 to get F(x):::; liminfFn(X):::; limsupFn(x):::; F(x). n-+00 n-+oo 0 The converse if false: Recall Example 8. 1.1. Despite the fact that convergence in distribution does not imply almost sure convergence, Skorohod's theorem provides a partial converse.
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8.3 The Baby Skorohod Theorem 259 Theorem 8.3.2 (Baby Skorohod Theorem) Suppose {Xn, n 0} are random variables defined on the probability space (Q, B, P) such that Then there exist random variables {X!, n 0} defined on the Lebesgue proba- bility space ([0, 1], 8([0, 1]), A = Lebesgue measure) such that for each fixed n 0, and X # a.s. X# n-+ 0 where a.s. means almost surely with respect to A. Note that Skorohod's theorem ignores dependencies in the original {Xn} se- quence. It produces a sequence {X!} whose one dimensional distributions match those of the original sequence but makes no attempt to match the finite dimen- sional distributions. The proof of Skorohod's theorem requires the following result. Lemma 8.3.1 Suppose Fn is the distribution function of X n so that Fn => Fo. If t e (0, 1) n C(F 0 -), then Fn<- (t) -+ F 0 <- (t). Proof of Lemma 8.3.1. Since C(Fo)c is at most countable, given E > 0, there exists x e C(Fo) such that From the definition of the inverse function, x < F 0 -(t) implies that Fo(x) < t.
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