Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices
Hock Peng Chan and Tze Leung Lai
Source: Ann. Probab. Volume 34, Number 1 (2006), 80-121.
Abstract
Several classical results on boundary crossing probabilities of Brownian motion 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 and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain “clumps” around high-level crossings. We also discuss how these results are related to the Poisson clumping heuristic and tube formulas of Gaussian random fields, and describe their applications to laws of the iterated logarithm in the form of the Kolmogorov–Erdős–Feller integral tests.
Primary Subjects: 60F10, 60G60
Secondary Subjects: 60F20, 60G15
Keywords: Multivariate empirical processes; moderate deviations; random fields; integral tests; boundary crossing probability
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1140191533
Digital Object Identifier: doi:10.1214/009117905000000378
Mathematical Reviews number (MathSciNet):
MR2206343
Zentralblatt MATH identifier:
05031260
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