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 , where *e* is a Brownian functional 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: . (The 's are independent variables with mean zero, is a martingale, and .) We further show that these inequalities are false if *t* or *n* is a stopping time, or if .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0312566-3

Keywords:
Martingale transform,
martingale,
stochastic integral,
convexity,
convex function,
inequalities

Article copyright:
© Copyright 1972
American Mathematical Society