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 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 ξ ₙ (t) converges weakly to the stochastic process ξ ₀ (t) if, for every finite aggregate t₁ ,t₂ , ⋯ ,tₘ, the sequence of the distributions of therandom vectors \ ξ ₙ (t₁ ), ⋯ ,ξ ₙ (tₘ )\ converges weakly to that of \ ξ ₀ (t₁ ), ⋯ ,ξ ₀ (tₘ )\. Theorem 2. Given a sequence of stochastic processes, $(t)$which converges weakly to a separable stochastic processξ 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 \[ { P}\{ g(t) ξ _n (t) f(t);{ for all }t\} [{n → ∞ }]{ P}\{ g(t) ξ _0 (t) f(t);{ for all }t\} . \] This theorem is used to study the asymptotic distributions of certain tests.