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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An extremal property of independent random variables


Authors: Steven Rosencrans and Stanley Sawyer
Journal: Proc. Amer. Math. Soc. 36 (1972), 552-556
MSC: Primary 60G45; Secondary 60G50, 60J65
MathSciNet review: 0312566
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Abstract: In a previous paper the first author proved $ Ef(\smallint_0^t {e\,db) \leqq Ef(M{b_t})} $, where e is a Brownian functional $ \leqq M$ in absolute value and f is a convex function such that the right side is finite. We now prove a discrete analog of this inequality in which the integral is replaced by a martingale transform: $ Ef(\sum\nolimits_1^n {{d_k}{y_k}) \leqq Ef(M\sum\nolimits_1^n {{y_k})} } $. (The $ {y_j}$'s are independent variables with mean zero, $ j \to {d_1}{y_1} + \cdots + {d_j}{y_j}$ is a martingale, and $ 0 \leqq {d_j} \leqq M$.) We further show that these inequalities are false if t or n is a stopping time, or if $ {d_j} \ngtr 0$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0312566-3
PII: S 0002-9939(1972)0312566-3
Keywords: Martingale transform, martingale, stochastic integral, convexity, convex function, inequalities
Article copyright: © Copyright 1972 American Mathematical Society