Large Deviations for Boundary Crossing Probabilities

For random walks sₙ, n = 1,2, ⋯ whose distribution can be imbedded in an exponential family, a method is described for determining the asymptotic behavior as m → ∞ of Pₙ > m c(n/m) some n < m sₘ = m μ₀\, μ₀ < c(1). Applications are given to the distribution of the Smirnov statistic and to modified repeated significance tests.