Weak Convergence of Stochastic Processes Whose Trajectories Have No Discontinuities of the Second Kind and the “Heuristic” Approach to the Kolmogorov-Smirnov Tests

Abstract

Theorem 1. Given a separable stochastic process satisfying the inequality (2) — Cbeing a constant independent oft — the random function$x(t)$defined by this process has, with probability 1, no points o f discontinuity of the second kind. We shall say that the sequence of stochastic processes $\xi _n (t)$ converges weakly to the stochastic process $\xi _0 (t)$ if, for every finite aggregate $t_1 ,t_2 , \cdots ,t_m $, the sequence of the distributions of therandom vectors $\{ \xi _n (t_1 ), \cdots ,\xi _n (t_m )\} $ converges weakly to that of $\{ \xi _0 (t_1 ), \cdots ,\xi _0 (t_m )\} $. Theorem 2. Given a sequence of stochastic processes, $(t)$which converges weakly to a separable stochastic process$\xi _0 (t)$in such a way that conditions (17) and (18) are satisfied, where functions $g(t)$ and $f(t)$$(g(t)f(t))$ are supposed to be semicontinuous. Then \[ {\bf P}\left\{ g(t) \leqq \xi _n (t) \leqq f(t);{\text{ for all }}t\right\} \xrightarrow[{n \to \infty }]{\bf P}\left\{ g(t) \leqq \xi _0 (t) \leqq f(t);{\text{ for all }}t\right\} . \] This theorem is used to study the asymptotic distributions of certain tests.