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), 233-239

MSC:
Primary 60E15; Secondary 60G42, 60G50

DOI:
https://doi.org/10.1090/S0002-9939-1990-0990432-9

MathSciNet review:
990432

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Abstract | References | Similar Articles | Additional Information

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,

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 one-sided approximation to the order of magnitude of expectations of functions of martingales. This approximation is best possible among all approximations depending only on the one-dimensional distribution of the martingale differences.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-0990432-9

Keywords:
Martingales,
sums of independent random variables

Article copyright:
© Copyright 1990
American Mathematical Society