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

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
MathSciNet review: 990432
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Abstract: Let $ \left( {{x_i}} \right)$ be a sequence of random variables. Let $ \left( {{w_i}} \right)$ be a sequence of independent random variables such that for each $ i, {w_i}$, has the same distribution as $ {x_i}$. If $ {S_n} = {x_1} + {x_2} + \cdots + {x_n}$ is a martingale and $ \Psi$ is a convex increasing function such that $ \Psi \left( {\sqrt x } \right)$ is concave on $ [0,\infty )$ and $ \Psi (0) = 0$ then,

$\displaystyle E\Psi \left( {{{\max }_{j \leq n}}\left\vert {\sum\limits_{i = 1}... ...CE\Psi \left( {\left\vert {\sum\limits_{i = 1}^j {{w_i}} } \right\vert} \right)$

for a universal constant $ C,(0 < C < \infty )$ independent of $ \Psi ,n$, and $ \left( {{x_i}} \right)$.

The same inequality holds if $ \left( {{x_i}} \right)$ is a sequence of nonnegative random variables and $ \Psi $ is now any nondecreasing concave function on $ [0,\infty )$ with $ \Psi (0) = 0$. Interestingly, if $ \Psi \left( {\sqrt x } \right)$ is convex and $ \Psi$ 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: http://dx.doi.org/10.1090/S0002-9939-1990-0990432-9
PII: S 0002-9939(1990)0990432-9
Keywords: Martingales, sums of independent random variables
Article copyright: © Copyright 1990 American Mathematical Society