Tail Estimates for Empirical Characteristic Functions, with Applications to Random Arrays

Abstract

In this paper we present an upper bound for the tail of the distribution of the supremum (taken for t in a fixed set K) of the modulus of expressions of the form

$$G(t)=\sum_{n=1}^{n}a_{n}(e^{iX_{n}t}-E[e^{iX_{n}t}])$$

where X ₁, X ₂,…, X _N are independent random variables each of which assumes values in some bounded interval [—L, L], a ₁,a ₂,…,a _n are real or complex constants, N is a fixed positive integer, t is a real number, and K is some interval in ℜ₁ of Lebesgue measure ∣K∣. Our specific interest in this problem is in obtaining explicit and mathematically rigorous numerical bounds for probabilities of the above type, with the goal of applying our results to response patterns that arise from arrays of randomly placed sensing devices. Thus, in addition to providing the probabilistic bounds described above we devote part of this paper to an explanation of a particular setting to which they apply. We comment at the outset that the problem of the distribution of the supremum of expressions (1.1) has been investigated in both the mathematical and the engineering literature, but that the two disciplines have had somewhat different research objectives and methods. These separate histories are traced briefly below, for there has been relatively little interplay between them.