On Probability and Moment Inequalities for Supermartingales and Martingales∗
نویسنده
چکیده
(Translated by the author) Abstract. The probability inequality for maxk≤n Sk, where Sk = ∑k j=1 Xj , is proved under the assumption that the sequence Sk, k = 1, . . . , n is a supermartingale. This inequality is stated in terms of probabilities P(Xj > y) and conditional variances of random variables Xj , j = 1, . . . , n. As a simple consequence the well-known moment inequality due to Burkholder is deduced. Numerical bounds are given for constants in Burkholder’s inequality.
منابع مشابه
Strong supermartingales and limits of non - negative martingales ∗
Given a sequence (M)n=1 of non-negative martingales starting at M n 0 = 1 we find a sequence of convex combinations (M̃)n=1 and a limiting process X such that (M̃n τ ) ∞ n=1 converges in probability to Xτ , for all finite stopping times τ . The limiting process X then is an optional strong supermartingale. A counter-example reveals that the convergence in probability cannot be replaced by almost ...
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