Invariance Principles and Self-Normalizations for Sums Trimmed According to Choice of Influence Function

Abstract

Asymptotic normality for sums of independent random variables has numerous useful applications in probability and statistics. This paper considers ways of modifying the summands of sums of i.i.d. random variables in order to get invariance principles when the random variables fail to have second or even first moments. Asymptotic normality fails for sums of i.i.d. random variables principally because of terms of large magnitude (Lévy 1937). So conceivably central limit theorems might be obtainable if these terms of large magnitudes are moderated or discounted. The inspiration for this work comes from three papers by Hahn and Kuelbs (1988) and Hahn, Kuelbs and Weiner (1990a,b) which investigate universal asymptotic normality. Respectively the truncating function T t (x) = xI(|x| ≤ t) and the censoring function C t (x) = (|x| ˄t)sgn(x) were applied to the sample to prove two universal type central limit theorems for totally modified and conditionally modified sums, and their empirical versions. Our goal is to obtain invariance principles for a large class of functions H t (x) which includes those for which T t (x) ≤ H t (x)≤ C t (x). These functions H t might be called influence functions because they determine what influence the large terms have on the partial sums. The approach here is necessariy different.