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

Theorem 1. Given a separable stochastic process satisfying the inequality (2) — Cbeing a constant independent oft — the random functionx (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 ₙ (t) converges weakly to the stochastic process ₀ (t) if, for every finite aggregate t₁, t₂, , tₘ, the sequence of the distributions of therandom vectors \ ₙ (t₁), , ₙ (tₘ) \ converges weakly to that of \ ₀ (t₁), , ₀ (tₘ) \. Theorem 2. Given a sequence of stochastic processes, (t) which converges weakly to a separable stochastic process 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 \ P\ g (t) ₙ (t) f (t) ;{ for all t\} [n P\ g (t) ₀ (t) f (t) ;{ for all t\}. \] This theorem is used to study the asymptotic distributions of certain tests.