An Extension of the Kolmogorov Distribution

Abstract

Let $x_1, x_2, \cdots, x_n, x'_1, x'_2, \cdots, x'_{nk}$ be independent random variables with a common continuous distribution $F(x)$. Let $x_1, x_2, \cdots, x_n$ have the empiric distribution $F_n(x)$ and $x'_1, x'_2, \cdots, x'_{kn}$ have the empiric distribution $G_{nk}(x)$. The exact values of $P(-y < F_n(s) - G_{nk}(s) < x$ for all $s$) and $P(-y < F(s) - F_n(s) < x$ for all $s$) are obtained, as well as the first two terms of the asymptotic series for large $n$.