Approximate Tail Probabilities for the Maxima of Some Random Fields

For random walks \Sₙ\ whose distribution can be embedded in an exponential family, large-deviation approximations are obtained for the probability that ₀ ₈ < ₉ ₌ (Sⱼ - Sᵢ) b (i) conditionally given Sₘ 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.