Tail Approximations for Maxima of Random Fields by Likelihood Ratio Transformations

The likelihood ratio method for dealing with change-point problems of Yakir and Pollak (1998 Yakir , B. and Pollak , M. ( 1998 ). A New Representation for a Renewal-Theoretic Constant Appearing in Asymptotic Approximations of Large Deviations , Annals of Applied Probability 8 : 749 – 774 .[Crossref], [Web of Science ®] , [Google Scholar]), which has subsequently been extended to deal with a wide variety of problems involving maxima of random fields, has as a key ingredient a conditional local limit theorem for a log-likelihood ratio, given an almost independent “local” sigma-algebra. This article contains a general version of that theorem, illustrated by several examples.