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The theory of boundary crossing probabilities in the study of repeated likelihood ratio tests was developed by Lai, Siegmund and Woodroofe in a series of articles and monographs appearing in the late seventies and early to mid eighties. This form part of the foundation for subsequent developments in the analysis of maxima of Gaussian and Poisson random fields used to provide accurate tail probability approximations of scan statistics. In this paper, we (i) track these theoretical developments,doi:10.1007/978-0-8176-4749-0_4 fatcat:ep2o3det6bdhdhv2z5nd2lakcm