Necessary and sufficient conditions of weak convergence of stochastic processes

Abstract

In this case we shall say that an s.p. {~ (t), t ~ T} is defined in (x, ~). Let an extended sequence of s. p.'s {~a (t), t T}, c~ ~ A according to some directed set A and an s. p. { ~ (t),t ~ T} be defined in (x, ~). We denote the induced measures on (x, e), corresponding to the s. p.'s just mentioned by Pa, a ~ A, and ~, respectively. The weak convergence of the s.p. ~ ~ ~, c~ ~ A is determined as the weak convergence of the induced measures ~a =~ P, c~ ~ A, i.e., as the fulfillment of the following limiting relation: for all f ~ C O (~), where C O ($) is the set of bounded and continuous complex-valued functionais on ~. We denote by ~ the space of the probability measures on (~, e), provided with a topology of weak convergence. In applications of theorems on weak convergence ~ o~ ~ ~, ~ ~ A, it is highly convenient to verify the condition, expressed in terms of the convergence of finite-dimension al distributions of s. p.'s { ~ (t), t ~ T}, ~ A. Precisely speaking, we have, on account of the following two conditions, which may be sufficient or necessary and sufficient for the convergence ~c~ ~ ~, c~ ~ A : 1 ~ for an arbitrary finite set T, ~ T joint distributions {~c~(t), t ~ T'}, c~ ~A converge to the distribution { ~ (t), t ~ T'} ; 2~ from each partial sequence {p ~,}a,e A ~ {U ~}a ~ A it is possible to extract a sequence converging in