On a non-classical invariance principle

We consider the invariance principle without the classical condition of asymptotic negligibility of individual terms. More precisely, let r.v.'s {[xi]nj} and {[eta]nj} be such that and the r.v.'s {[eta]nj} are normal. We set Let Xn(t) and Yn(t) be continuous piecewise linear (or polygonal) random functions with vertices at (tkn,Skn) and (tkn,Ykn), respectively, and let Pn and Qn be the respective distributions of the processes Xn(t) and Yn(t) in . The goal of the present paper is to establish necessary and sufficient conditions for convergence of Pn-Qn to zero measure not involving the condition of the asymptotic negligibility of the r.v.'s {[xi]nj} and {[eta]nj}.