By William F. Stout

ISBN-10: 0126727503

ISBN-13: 9780126727500

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Clearly EUt = 1 yet EUX = 0. Interestingly EUil) = 0 for all n > 1. 2. s. for some M < oo. Prove that E t Yt = 0. ) | Besides the optional stopping discussed above, optional sampling of a martingale (or submartingale) is possible: Let 1 < tx < t2 < • • • < tn < • • • be a sequence of stopping rules and {Un, n > 1} a martingale. Does it follow that Utl, Ut2, . . , Utn, . . is also a martingale? The coin 40 POINTWISE CO N VERGEN CE OF PARTIAL SUMS Chapter 2 tossing example just discussed shows that the answer in general is no.

1. Proof. 1. | If {Un , n > 1} is a martingale with Un representing a gambler’s ac­ cumulated winnings at trial n of a nonending game, we might wish to build into the mathematical structure the possibility of the gambler stopping play at some finite time t with winnings Ut . This optional stopping preserves the martingale property in a way made precise below. Let 1} be an increasing sequence of o fields with ^ ci ^ for each n > 1. A function f, defined on Q, is called a stopping rule if t has range {1, 2, .

Ii) If 0 is a continuous convex real valued function defined on and {Un, n > 1} is a martingale with E \ 0{UV) | < oo for each n > 1, then {0(Un), n > 1} is a submartingale. 1. 6 PRESERVATION OF THE MARTINGALE STRUCTURE 37 (i) Fix n > 2 . s. Using the submartingale assumption and the fact that 0 is nonde­ creasing, the desired result follows. The proof of (ii) is essentially the same and omitted. | Proof, (i) Fix p > 1. If {C/n, n > 1} is a martingale with E \ Un \* < oo for n > 1 then {| Un ¡p, n > 1} is a submartingale.

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Almost Sure Convergence by William F. Stout

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