# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Bounds on the expectation of functions of martingales and sums of positive RVs in terms of norms of sums of independent random variablesHTML articles powered by AMS MathViewer

by Victor H. de la Peña
Proc. Amer. Math. Soc. 108 (1990), 233-239 Request permission

## 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, $E\Psi \left ( {{{\max }_{j \leq n}}\left | {\sum \limits _{i = 1}^j {{x_i}} } \right |} \right ) < CE\Psi \left ( {\left | {\sum \limits _{i = 1}^j {{w_i}} } \right |} \right )$ 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.
References
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60E15, 60G42, 60G50
• Retrieve articles in all journals with MSC: 60E15, 60G42, 60G50