A Method of Approximating Expectations of Functions of Sums of Independent Random Variables

Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent random variables with $S_n = \sum^n_{i = 1} X_i$. Fix $\alpha > 0$. Let $\Phi(\cdot)$ be a continuous, strictly increasing function on $\lbrack 0, \infty)$ such that $\Phi(0) = 0$ and $\Phi(cx) \leq c^\alpha\Phi(x)$ for all $x > 0$ and all $c \geq 2$. Suppose $a$ is a real number and $J$ is a finite nonempty subset of the positive integers. In this paper we are interested in approximating $E \max_{j \in J} \Phi(|a + S_j|)$. We construct a number $b_J(a)$ from the one-dimensional distributions of the $X$'s such that the ratio $E \max_{j \in J} \Phi(|a + S_j|)/\Phi(b_J(a))$ is bounded above and below by positive constants which depend only on $\alpha$. Bounds for these constants are given.