An Extension of the Kolmogorov Distribution

Let x₁, x₂, , xₙ, x'₁, x'₂, , x'₍₊ be independent random variables with a common continuous distribution F (x). Let x₁, x₂, , xₙ have the empiric distribution Fₙ (x) and x'₁, x'₂, , x'₊₍ have the empiric distribution G₍₊ (x). The exact values of P (-y < Fₙ (s) - G₍₊ (s) < x for all s) and P (-y < F (s) - Fₙ (s) < x for all s) are obtained, as well as the first two terms of the asymptotic series for large n.