4.4 Doob s inequalities

Transcription

1 34 CHAPTER 4. MARTINGALES 4.4 Doob s nequaltes The frst nterestng consequences of the optonal stoppng theorems are Doob s nequaltes. If M n s a martngale, denote M n =max applen M. Theorem 4.8 If M n s a martngale or a postve submartngale, P(M n a) apple E [ M n ; M n a]/a apple E M n /a. Proof. Set M n+1 = M n.letn =mn{ : M a}^(n +1). Snce s convex, M n s a submartngale. If A =(Mn a), then A 2F N because A \ (N apple ) =(N apple n) \ (N apple ) =(N apple ) 2F. By Corollary 4.4 P(M n a) apple E h M n a ; M n a apple 1 a E [ M N ; A] apple 1 a E [ M n ; A] apple 1 a E M n. For p>1, we have the followng nequalty. Theorem 4.9 If p>1 and E M p < 1 for apple n, then p pe E (Mn) p apple Mn p. p 1 Proof. Note Mn apple P n =1 M n, hencemn 2 L p. We wrte, usng Theorem 4.8 for the frst nequalty, E (M n) p = = E apple Z 1 pa p Z M n 1 P(M n >a) da apple Z 1 pa p 1 E [ M n 1 (M n a)/a] da pa p 2 M n da = p p 1 E [(M n) p 1 M n ] p p 1 (E (M n) p ) (p 1)/p (E M n p ) 1/p. The last nequalty follows by Hölder s nequalty. Now dvde both sdes by the quantty (E (M n) p ) (p 1)/p.

2 4.5. MARTINGALE CONVERGENCE THEOREMS Martngale convergence theorems The martngale convergence theorems are another set of mportant consequences of optonal stoppng. The man step s the upcrossng lemma. The number of upcrossngs of an nterval [a, b] sthenumberoftmesaprocess crosses from below a to above b. and To be more exact, let S 1 =mn{k : X k apple a}, T 1 =mn{k >S 1 : X k b}, S +1 =mn{k >T : X k apple a}, T +1 =mn{k >S +1 : X k b}. The number of upcrossngs U n before tme n s U n =max{ : T apple n}. Theorem 4.1 (Upcrossng lemma) If X k s a submartngale, E U n apple (b a) 1 E [(X n a) + ]. Proof. The number of upcrossngs of [a, b] byx k s the same as the number of upcrossngs of [,b a] byy k =(X k a) +.MoreoverY k s stll a submartngale. If we obtan the nequalty for the the number of upcrossngs of the nterval [,b a] bytheprocessy k,wewllhavethedesrednequalty for upcrossngs of X. So we may assume a =. Fx n and defne Y n+1 = Y n. Ths wll stll be a submartngale. Defne the S, T as above, and let S = S ^ (n +1), T = T ^ (n +1). SnceT +1 >S +1 >T,thenTn+1 = n +1. We wrte Xn+1 Xn+1 E Y n+1 = E 1 + E [Y T ]+ E [ +1 Y T ]. = All the summands n the thrd term on the rght are nonnegatve snce Y k s asubmartngale. Fortheth upcrossng, Y T b s always greater than or equal to. So a, whley T (Y T ) (b a)u n. = =

3 36 CHAPTER 4. MARTINGALES So E U n apple E Y n+1 /(b a). (4.2) Ths leads to the martngale convergence theorem. Theorem 4.11 If X n s a submartngale such that sup n E X n + X n converges a.s. as n!1. < 1, then Proof. Let U(a, b) =lm n!1 U n. For each a, b ratonal, by monotone convergence, E U(a, b) apple c(b a) 1 E (X n a) + < 1. So U(a, b) < 1, a.s. Takng the unon over all pars of ratonals a, b, wesee that a.s. the sequence X n (!) cannothavelmsupx n > lm nf X n.therefore X n converges a.s., although we stll have to rule out the possblty of the lmt beng nfnte. Snce X n s a submartngale, E X n E X,andthus E X n = E X + n + E X n =2E X + n E X n apple 2E X + n E X. By Fatou s lemma, E lm n X n applesup n E X n < 1, orx n converges a.s. to afntelmt. Corollary 4.12 If X n s a postve supermartngale or a martngale bounded above or below, X n converges a.s. Proof. If X n s a postve supermartngale, X n s a submartngale bounded above by. Now apply Theorem If X n s a martngale bounded above, by consderng X n,wemayassume X n s bounded below. Lookng at X n + M for fxed M wll not a ect the convergence, so we may assume X n s bounded below by. Now apply the frst asserton of the corollary.

4 4.6. APPLICATIONS OF MARTINGALES 37 Proposton 4.13 If X n s a martngale wth sup n E X n p < 1 for some p>1, then the convergence s n L p as well as a.s. Ths s also true when X n s a submartngale. If X n s a unformly ntegrable martngale, then the convergence s n L 1.IfX n! X 1 n L 1, then X n = E [X 1 F n ]. X n s a unformly ntegrable martngale f the collecton of random varables X n s unformly ntegrable. Proof. The L p convergence asserton follows by usng Doob s nequalty (Theorem 4.9) and domnated convergence. The L 1 convergence asserton follows snce a.s. convergence together wth unform ntegrablty mples L 1 convergence. Fnally, f <n,wehavex = E [X n F ]. If A 2F, E [X ; A] =E [X n ; A]! E [X 1 ; A] by the L 1 convergence of X n to X 1. Snce ths s true for all A 2 F, X = E [X 1 F ]. 4.6 Applcatons of martngales One applcaton of martngale technques s Wald s denttes. Proposton 4.14 Suppose the Y are..d. wth E Y 1 < 1, N s a stoppng tme wth E N < 1, and N s ndependent of the Y. Then E S N = (E N)(EY 1 ), where the S n are the partal sums of the Y. Proof. S n n(e Y 1 )samartngale,soes n^n = E (n ^ N)E Y 1 by optonal stoppng. The rght hand sde tends to (E N)(E Y 1 )bymonotone convergence. S n^n converges almost surely to S N,andweneedtoshowthe expected values converge. Note S n^n = = S n^k 1 (N=k) apple k= nx = k> Y 1 (N=k) = Xn^k Y 1 (N=k) k= = nx Y 1 (N ) apple = Y 1 (N ). =

5 38 CHAPTER 4. MARTINGALES The last expresson, usng the ndependence, has expected value (E Y )P(N ) apple (E Y 1 )(1 + E N) < 1. = So by domnated convergence, we have E S n^n! E S N. Wald s second dentty s a smlar expresson for the varance of S N.