New PDF release: Almost Sure Convergence

By William F. Stout

ISBN-10: 0126727503

ISBN-13: 9780126727500

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E(Hy ) < E[(UN - a)+]/(b - a). s. By Eq. 1), t W * - ^ - i ) < (« - b)Hs + (C/A- - a)+. Ik is < k — 1) measurable and hence observation. s. for k > 2. Taking expectations in Eq. 3) implies that (a - b)EHN + E[(Un - u)+] > 0 and hence that E(Hy ) < E[(Un - a)+]/(b - a). | It is now an easy matter to derive the submartingale convergence theo­ rem. This result is without doubt the most important martingale convergence result, as demonstrated by its wide applicability. [Doob, 1953, p. 319]. s. and E \ S | < sup E | Sn |.

It is left to the reader to provide a formal proof by computing the ap­ propriate conditional expectations. 2 to yield a simple martingale proof of an interesting result in the theory of Haar series: [Talalyan and Arutyunyan, 1965]. s. cannot converge to +oo (or —oo) on a set of positive Lebesgue measure. 4. R em ark. s. See p. 91 for a martingale proof of this deeper result. P roof. Suppose that a Haar series X S i a%^% does converge to + oo on a set A of positive Lebesgue measure. 2 that £ £ i converges on A, a contradiction.

1. ). Let { ^ ¿ , / > 1 } be a sequence of in­ dependent identically distributed random variables with P[Xx = 1] + P[Xx = - 1 ] = 1 and P[Xx = 1] < i (that is, coin tossing random variables with the coin possibly unfair). Let 46 POINTWISE CO N VERGEN CE OF PARTIAL SUMS Chapter 2 at be ¿%(X1, . . , Xi-j) measurable for each i > 2 and constant. For each / > 1, aiXi represents the gambler’s winnings on trial /, given that the gambler bets on trial z. Let n Tn = Y j aiXi for each n > 1 and T0 = 1 i= l (that is, the gambler’s initial fortune is 1).

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Almost Sure Convergence by William F. Stout


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