Bounds on the expectation of functions of martingales and sums of positive RVs in terms of norms of sums of independent random variables
Author:
Victor H. de la Peña
Journal:
Proc. Amer. Math. Soc. 108 (1990), 233239
MSC:
Primary 60E15; Secondary 60G42, 60G50
MathSciNet review:
990432
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Abstract: Let be a sequence of random variables. Let be a sequence of independent random variables such that for each , has the same distribution as . If is a martingale and is a convex increasing function such that is concave on and then, for a universal constant independent of , and . The same inequality holds if is a sequence of nonnegative random variables and is now any nondecreasing concave function on with . Interestingly, if is convex and grows at most polynomially fast, the above inequality reverses. By comparing martingales to sums of independent random variables, this paper presents a onesided approximation to the order of magnitude of expectations of functions of martingales. This approximation is best possible among all approximations depending only on the onedimensional distribution of the martingale differences.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199009904329
PII:
S 00029939(1990)09904329
Keywords:
Martingales,
sums of independent random variables
Article copyright:
© Copyright 1990 American Mathematical Society
