On a non-classical invariance principle

Abstract

We consider the invariance principle without the classical condition of asymptotic negligibility of individual terms. More precisely, we explore the difference of the following two distributions in the space C (of continuous functions on [0,1]). The first is the distribution of the continuous piecewise linear partial-sum process generated by a sequence of independent random variables, and the second is the distribution of the similar process generated by the sequence of normal r.v.'s with the same first two moments. The novelty is that the condition of negligibility of the r.v.'s is not imposed. We establish a necessary and sufficient condition of the weak convergence of the difference mentioned to zero measure in C.