Approximate Tail Probabilities for the Maxima of Some Random Fields

For random walks $\{S_n\}$ whose distribution can be embedded in an exponential family, large-deviation approximations are obtained for the probability that $\max_{0\leq i < j\leq m}(S_j - S_i) \geq b$ (i) conditionally given $S_m$ and (ii) unconditionally. The method used in the conditional case seems applicable to maxima of a reasonably large class of random fields. For the unconditional probability a more special argument is used, and more precise results obtained.