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 {ξnj}{ξnj} and {ηnj}{ηnj} be such that E{ξnj}=E{ηnj}=0,E{ξnj2}=E{ηnj2}=σnj2,∑jσnj2=1, and the r.v.’s {ηnj}{ηnj} are normal. We set Skn=∑j=1kξnj,Ykn=∑j=1kηnj,tkn=∑j=1kσnj2. Let Xn(t)Xn(t) and Yn(t)Yn(t) be continuous piecewise linear (or polygonal) random functions with vertices at (tkn,Skn)(tkn,Skn) and (tkn,Ykn)(tkn,Ykn), respectively, and let PnPn and QnQn be the respective distributions of the processes Xn(t)Xn(t) and Yn(t)Yn(t) in C[0,1]C[0,1]. The goal of the present paper is to establish necessary and sufficient conditions for convergence of Pn−QnPn−Qn to zero measure not involving the condition of the asymptotic negligibility of the r.v.’s {ξnj}{ξnj} and {ηnj}{ηnj}.