Note a says that for any s there is no subsequential

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Note (a) says that for any s, there is no subsequential limit bigger than 1 +sand (b) says that for any s, there is always some subsequential limit bounded below by 1- s. We have the following set equality: Let Sk .J.. 0 and observe [ . En ) hmsup-- = 1 n-+00 logn =n{liminf[En :;:1+sk]}nn{[ 1 En >1-sk]i .o.} (4.16) k n-+oo logn k ogn To prove that the event on the left side of (4.16) has probability 1, it suffices to prove every braced event on the right side of (4.16) has probability 1. For fixed k ""' En ""' L... P[- > 1- Sk] = L... P[En > (1- Sk) logn] n logn n = I:exp{-(1-Ek)logn} n 1 = L nl - £k = oo . n So the Borel Zero-One Law 4.5.2 implies Likewise so P { [ > 1 - Ek] i.o. } = 1. logn ""' En ""' L... P[-- > 1 + sk] = L... exp{-(1 + Sk) logn} n logn n 1 = L nl+£k < oo, n P (lim sup [ En > 1 + Sk]) = 0 n-+oo logn implies P =:: 1 + Sk]} = 1- P {lim sup [ En :;: 1 + sk]c} = 1. n-+00 logn n-+00 logn 0
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4.5 Independence, Zero-One Laws, Borel - Cantelli Lemma 107 4.5.3 Kolmogorov Zero-One Law Let {X n} be a sequence of random variables and define = a(Xn+l• Xn+2• . .. ), n = 1, 2, .... The tail a-field Tis defined as lim ,l.a(Xn.Xn+J. ... ). n-+00 n These are events which depend on the tail of the {X n } sequence . If A e T, we will call A a tail event and similarly a random variable measurable with respect to T is called a tail random variable. We now give some examples of tail events and random variables. 1. Observe that 00 {w : LXn(w) converges} E T. n=l To see this note that, for any m, the sum Xn(w) converges if and only if I::m Xn(w) converges. So 00 [LX n converges ] = [ L X n converges ] e n n=m+l This holds for all m and after intersecting over m. 2. Wehave limsupXn E T, n-+oo {w : lim Xn(w) exists} E T. n-+oo This is true since the lim sup of the sequence {X t. X 2, ... } is the same as the lim sup of the sequence {X m , X m+l , ... } for all m. 3. Let Sn = X 1 + · · · + X n. Then { w: lim Sn(w) =0} eT n-+oo n since for any m, ll ·m Sn(w) __ 1 . L7= 1 X;(w) 1 . L7=m+1X;(w) Im = Im ' n-+oo n n-+oo n n-+oo n and so for any m,
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108 4. Independence Call a a-field, all of whose events have probability 0 or 1 almost trivial. One example of an almost trivial a-field is the a-field {0, Q}. Kolmogorov 's Zero-One Law characterizes tail events and random variables of independent sequences as almost trivial. Theorem 4.5.3 (Kolmogorov Zero-One Law) If (Xn} are independent random variables with tail a-field T, then A E T implies P(A) = 0 or 1 so that the tail a-field Tis almost trivial. Before proving Theorem 4.5. 3, we consider some implications. To help us do this, we need the following lemma which provides further information on almost trivial a-fields. Lemma 4.5.1 (Almost trivial a-fields) Let g be an almost trivial a- field and let X be a random variable measurable with respect to g. Then there exists c such that P[X = c] = 1. Proof of Lemma 4.5.1. Let F(x) = P[X:::;: x]. Then F is non-decreasing and since [X:::;: x] e a(X) C g, F(x) = 0 or 1 for each x e JR. Let c = sup{x : F(x) = 0}. The distribution function must have a jump of size 1 at c and thus P[X = c] = 1.
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