Large Deviations for Boundary Crossing Probabilities

For random walks $s_n, n = 1,2, \cdots$ whose distribution can be imbedded in an exponential family, a method is described for determining the asymptotic behavior as $m \rightarrow \infty$ of $P\{s_n > m c(n/m) \quad\text{for some}\quad n < m\mid s_m = m \mu_0\}, \quad\mu_0 < c(1).$ Applications are given to the distribution of the Smirnov statistic and to modified repeated significance tests.