# A second course in stochastic processes by Samuel Karlin

By Samuel Karlin

This moment direction keeps the advance of the idea and purposes of stochastic approaches as promised within the preface of a primary direction. We emphasize a cautious therapy of simple constructions in stochastic tactics in symbiosis with the research of average periods of stochastic procedures coming up from the organic, actual, and social sciences.

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Proof. The first assertion follows from Corollary 1 and the second from the inequality Xf < E(X^\&~T\ which is true for all accessible stopping points T, by Corollary 1. Theorem 2 (Closing a supermartingale indexed by 0 on the right). Let X be a supermartingale indexed by i whose negative part X~ is uniformly integrable. Then X, converges in probability when t -» ixi to an integrable random variable I K and (X,, t e I) is a supermartingale. Proof. Suppose that X, does not converge in probability when / -» XI .

D), we still have to check that sup, E s < ( Xs/(s)\ < oo, or equivalently, that sup, |E, s f (Y, - E(Ys))/(s)\ < oo. 14), we see that 2 / _ Yt - E(YS) \ 1 < -|sup E = -|sup ' DE £ Y, ~ E(V,) (s) y, - E ( y j JS/ 4EE i / y , - E ( y , ) <Í> *7? r«v supE(|^|")E From this, we conclude that £„ P(Fn) < °°, therefore, by the Borel-Cantelli lemma, that sup, |E iS ,(Y s - E(F s ))/ 1, then E(sup,(|5,|/

Indeed, if T e A and r is a predictable increasing path that passes through T, then Tn = T A Tn e Ab for all n e N and lim n _„ XT l { r < M} = A^ l { r < M} in L1, because Tn = T on {T < N} as soon as n is large enough, and the sequence (XT , n e N) is uniformly integrable by hypothesis. 2, this closure is uniformly integrable. Omitting the terms involving XK handles the case where the index set is I. 11) for all couples (s, t) of elements of J such that s < t. 11) be satisfied by the couples (s, t) such that t e Ds.