Bounds on the expectation of functions of martingales and sums of positive RVs in terms of norms of sums of independent random variables

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.