Approximate tail probabilities for the maxima of some random fields (Q1103951)

Let \(x_ 1\), \(x_ 2\),... be i.i.d. random variables, and put \(S_ n=x_ 1+...+x_ n\), \(n\geq 1\), \(S_ 0=0\). For \(b>0\) a large-deviation approximation to the conditional probability \[ P(\max_{0\leq i<j\leq m}(S_ j-S_ i)\geq b| S_ m) \] is given when the distribution of the x's can be embedded in an exponential family. The method used is a modification of that one developed in the one-dimensional parameter case by \textit{M. Woodroofe} [CBMS-NSF Reg. Conf. Ser. Appl. Math. 39 (1982; Zbl 0487.62062)]. For the unconditional probability a more special argument is used, and more precise results are obtained. The method used in the conditional case is shown to be applicable to maxima of a reasonably large class of random fields.