Thus n pisni a l pisn sjl a j j jl it is also true

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Thus, N P[ISNI >a]:=: L P[ISN- Sjl::: a, J = j]. j=l It is also true that and Since N SN- Sj = L Xj E B(Xj+l· ... ,XN) i=j+l [J = j] =[sup IS; I::: 2a, ISjl > 2a] E B(Xt ... Xj). i<j we have N P[ISN I > a] :=: L P[ISN - Sj I ::: a ]P[J = j] j=l N :=: L(l- c)P[J = j] (from the definition of c) j=l = (1 - c)P[J ::: N] = (1- c)P[sup ISjl > 2a]. j5N 0 Based on Skorohod's inequality, we may now present a rather remarkable result due to Levy which shows the equivalence of convergence in probability to almost sure convergence for sums of independent random variables. Reminder: If is a monotone sequence of random variables, then p implies (and hence is equivalent to) l: a.s. l:
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7.3 Almost Sure Convergence of Sums of Independent Random Variables 211 Theorem 7.3.2 (Uvy's theorem) If {Xn. n :::: 1} is an independent sequence of random variables, LX n converges i.p. iff LX n converges a.s. n n This means that if Sn = L:?=t X;, then the following are equivalent: 1. { Sn} is Cauchy in probability. 2. {Sn} converges in probability. 3. {Sn} converges almost surely. 4. {Sn} is almost surely Cauchy. Proof. Assume { Sn} is convergent in probability, so that { Sn} is Cauchy in proba- bility. We show { Sn} is almost surely convergent by showing { Sn} is almost surely Cauchy. To show that {Sn} is almost surely Cauchy, we need to show = sup ISm - Sn I --+ 0 a.s., m,n?;N as N --+ oo. But N :::: 1} is a decreasing sequence so from the reminder it suffices to show ..!:. 0 as N --+ oo . Since = sup ISm - SN + SN - Sn I m,n?;N it suffices to show that sup ISm- SNI +sup ISn- SNI m?;N n?;N =2 sup ISn- SNI n?;N =2sup ISN+j- SNI. j?;O (7.10) For any > 0, and 0 < < ! , the assumption that { Sn} is cauchy i. p. implies that there exists ,o such that (7.11) if m, m' :=::: N£,o• and hence (7.12)
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212 7. Laws of Large Numbers and Sums of Independent Random Variables if N ::: Nf,8· Now write = lim P[ sup ISN+j- SNI > E]. N'-+oo N'?.j:::.O Now we seek to apply Skorohod's inequality. Let x; = XN+i and j j sj = L:x; = L:xN+i = sN+j- sN. i=l i=l With this notation we have P[ sup ISN+j- SNI > E] N'?.j?.O from the choice of 8. Note that from (7.11) Since 8 can be chosen arbitrarily small, this proves the result. 0 Levy's theorem gives us an easy proof of the Kolmogorov convergence crite- rion which provides the easiest method of testing when a series of independent random variables with finite variances converges; namely by checking conver- gence of the sum of the variances. Theorem 7.3.3 (Kolmogorov Convergence Criterion) Suppose {Xn, n ::: 1} is a sequence of independent random variables. If then 00 00 L Var(Xj) < oo, j=l L (X j - E (X j)) converges almost surely. j=l
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7.4 Strong Laws of Large Numbers 213 Proof. Without loss of generality, we may suppose for convenience that E (Xi) = 0. The sum of the variances then becomes 00 LEX]< 00 . j=l This implies that {Sn} is Lz Cauchy since (m < n) n IISn- = Var(Sn- Sm) = L EXJ--+ 0, j=m+l as m, n--+ oo since Lj E(XJ) < oo. So {Sn}. being Lz-Cauchy, is also Cauchy in probability since n P[ISn- Sml > ] :S E- 2 Var(Sn- Sm) = E- 2 L Var(Xj)--+ 0 j=m as n, m --+ oo . By Levy's theorem {Sn} is almost surely convergent. D Remark. The series in Theorem 7.3.3 is Lz-convergent. Call the Lz limit L:j: 1 (Xj- E(Xj)).
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