Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices

摘要

Several classical results on boundary crossing probabilities of Brownianmotion and random walks are extended to asymptotically Gaussian random fields,which include sums of i.i.d. random variables with multidimensional indices,multivariate empirical processes, and scan statistics in change-point andsignal detection as special cases. Some key ingredients in these extensions aremoderate deviation approximations to marginal tail probabilities and weakconvergence of the conditional distributions of certain ``clumps'' aroundhigh-level crossings. We also discuss how these results are related to thePoisson clumping heuristic and tube formulas of Gaussian random fields, anddescribe their applications to laws of the iterated logarithm in the form ofthe Kolmogorov--Erd\H{o}s--Feller integral tests.