On excursion sets, tube formulas and maxima of random fields

This is a rambling review of what, with a few notable and significant exceptions, has been a rather dormant area for over a decade. It concentrates on the septuagenarian problem of finding good approximations for the excursion probability P{sup(t is an element ofT) X-t greater than or equal to lambda}, where lambda is large, X is a Gaussian, or "Gaussian-like," process over a region T subset of R-N and, generally, N > 1. A quarter of a century ago, there was a flurry of papers out of various schools linking this problem to the geometrical properties of random field sample paths. My own papers made the link via Euler characteristics of the excursion sets {t is an element of T: X-t greater than or equal to lambda}. A decade ago, Aldous popularized the Poisson clumping heuristic for computing excursion probabilities in a wide variety of scenarios, including the Gaussian. Over the past few years, Keith Worsley has been the driving force behind the computation of many new Euler characteristic functionals, primarily driven by applications in medical imaging. There has also been a parallel development of techniques in the astrophysical literature. Meanwhile, somewhat closer to home, Hotelling's 1939 "tube formulas" have seen a renaissance as sophisticated statistical hypothesis testing problems led to their reapplication toward computing excursion probabilities, and Sun and others have shown how to apply them in a purely Gaussian setting. The aim of the present paper is to look again at many of these results and tie them together in new ways to obtain a few new results and, hopefully, considerable new insight. The "Punchline of this paper," which relies heavily on a recent result of Piterbarg, is given in Section 6.6: "In computing excursion probabilities for smooth enough Gaussian random fields over reasonable enough regions, the expected Euler characteristic of the corresponding excursion sets gives an approximation, for large levels, that is accurate to as many terms as there are in its expansion."