A Probability Path.pdf

# Lx du fdx jo lxo uo 1 1 f fdx du f fudu tft uo lxu jo

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[lx du] F(dx) Jo lx=O u=O = 1 1 [ f' F(dx)] du = f' F(u)du - tF(t) u=O lx=u Jo ;::: lot F(u)du. From (7.21), given() > 0, there exists xo such that x 2: xo implies .F{x) ;::: (1 + ())kx-a =: k1x-a. Thus from (7 .26) 1 xo 1t 1' m(t);::: + ;:::c+kt u-adu, 0 xo xo t 2: XQ. Now for a> 1, E(IZII) < oo so that L iPiim(c/IPiD:::: L IPjiE(IZtD < oo j j by (7.23). For a = 1, we find from (7.28) that m(t) :::: c' + kzlogt, t::: xo (7.26) (7.27) (7.28) for positive constants c', kz. Now choose 1J > 0 so small that 1 - 1J > 8, and for another constant c" > 0 IPilm(c/IPiD:::: c" IPil + kz IPillog c:i,) 1 1 1 :::: c"L IPil +k3 L 1Pi1 1 - 11 < oo j j

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230 7. Laws of Large Numbers and Sums of Independent Random Variables where we have used (7.23) and Finally for a < 1, t > xo so that L IPjlm(l/IPjD :S Cz L IPjl +k4 L IPjll+a- 1 < oo j j j from (7.23). 0 7.6.1 Necessity of the Kolmogorov Three Series Theorem We now examine a proof of the necessity of the Kolmogorov three series theo- rem. Two lemmas pave the way. The first is a partial converse to the Kolmogorov convergence criterion. Lemma 7.6.1 Suppose {Xn, n ::: 1} are independent random variables which are uniformly bounded, so that for some a > 0 and all w e Q we have IX n (w) I ::: a. If"f:.n(Xn - E(Xn)) converges almost surely, then Var(Xn) < oo. Proof. Without loss of generality, we suppose E (X n) = 0 for all n and we prove the statement: If {Xn, n ::: 1} are independent, E(Xn) = 0, IXn I :Sa, then Ln Xn almost surely convergent implies Ln E (X;) < oo. We set Sn = "'£7= 1 X;, n ::: 1 and begin by estimating Var(SN) = "'f:.f: 1 E (X f) for a positive integer N. To help with this, fix a constant A > 0, and define the first passage time out of [-A, A] r := inf{n::: 1: ISnl >A}. Set r = oo on the set I ::: A]. We then have N LE(XI) = = + (7.29) i=1 =I+ II. Note on r > N, we have vf: 1 IS; I ::: A, so that in particular, ::: A 2 . Hence, (7.30) For I we have N I= L j=1
For j < N Note while and thus 7.6 The Kolmogorov Three Series Theorem 231 N = E((Sj + L X;) 2 )1[T=j]). i=j+l j-1 [r = j] = lV IS; I::: A, ISjl >A] E a(Xt. ... , Xj) i=1 N L X; E a(Xj+t. ... ,XN), i=j+l N l[T=j) JL L X;. i=j+1 Hence, for j < N, = E((SJ + 2Sj t X;+ ( t X;) 2 )l[T=j)) i=j+1 i=j+1 N = E(SJl[T=j]) + 2E(Sj1[T=jJ)E( L X;) i=j+1 N + E( L X;) 2 P[r = j] i=j+l N = E(SJl[T=j]) + 0 + E( L X;) 2 P[r = j] i=j+l N ::: E((ISj-11 + IXji) 2 1[T=j]) + L E(X;) 2 P[r = j] i=j+1 N _:::(A+a) 2 P[r=j]+ L E(X;) 2 P[r=j]. i=j+l Summarizing, we conclude for j < N that E(X;) 2 )P[r=j], (7.31) 1=]+1

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232 7. Laws of Large Numbers and Sums of Independent Random Variables and defining E(X;) 2 = 0, we find that (7.31) holds for 1 j N. Adding over j yields I= E(S11[r:::NJ) (o. +a) 2 + tE(Xf))P[r N] (7.32) Thus combining (7.30) and (7.32) N ECS1) = 'L,E<Xf) =I+ II i=l (o. + a) 2 + ECS1)) P[r N] +(A +a) 2 P[r > N] and solving for E (S1) yields Let N-+ oo. We get oo (A+ a)2 LE(Xf) ::S _ , i=l P[r- oo] which is helpful and gives the desired result only if P [ r = oo] > 0. However, note that we assume Ln X n is almost surely convergent, and hence for almost all w, we have {Sn(w), n :::: 1} is a bounded sequence of numbers. So VniSnl is almost surely a finite random variable, and there exists A > 0 such that 00 P[r = oo] = P[V ISnl 0, n=l else I < oo] = 0, which is a contradiction. This completes the proof of the lemma.
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