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 $P\{sup_{t \in T}X_t \geq \lambda\}$, where $\lambda$
is large, $X$ is a Gaussian, or “Gaussian-like,” process over a
region $T \subset \Re^N$, 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 $\{t \in T: X_t \geq
\lambda\}$. 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.”