## The Annals of Probability

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

#### 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.

#### Article information

Source
Ann. Probab., Volume 34, Number 1 (2006), 80-121.

Dates
First available in Project Euclid: 17 February 2006

https://projecteuclid.org/euclid.aop/1140191533

Digital Object Identifier
doi:10.1214/009117905000000378

Mathematical Reviews number (MathSciNet)
MR2206343

Zentralblatt MATH identifier
1106.60022

Subjects
Primary: 60F10: Large deviations 60G60: Random fields
Secondary: 60F20: Zero-one laws 60G15: Gaussian processes

#### Citation

Chan, Hock Peng; Lai, Tze Leung. Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices. Ann. Probab. 34 (2006), no. 1, 80--121. doi:10.1214/009117905000000378. https://projecteuclid.org/euclid.aop/1140191533

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