A limit theorem for measurable random processes and its applications

Let the measurable random processes ξ1(t)…., ξn(t)…. and ξ(l) be defined on [0, 1].There exists C such that for all n and t we have Eξn(t)p < C, p >1. The following assertion is valid: if for any finite set of points t1,…, tk ⊂ [0, 1] the joint distribution of ξ(t1),…, ξ(tk) convergesto the joint distribution of ξ(t1),……ξ(tk), and if Eξn(t)p ⇾ Eξ(t)p for all t ∊;[0, 1], then for any continuous functional ƒ on Lp[0, 1] thedistribution oƒ(ξ(t)) converges to the distribution of ƒ(ξ(t)).This statementimmediately implies the convergence of distributions in some limit theorems for the sums of independent random variables (for example, in oneof the theorems of P. Erdos and M. Kac) and in some statistical criteria (for example, in the ω2-criterion of Cramer and von Mises).