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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.
David Siegmund (Fri,) studied this question.