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MR2478201 (Review) 6002 (60E15 60G05 60G15 60G60 60G70)
Aza¨is, JeanMarc (FTOUL3SPM); Wschebor, Mario (URUREPSCM)
Level sets and extrema of random processes and fields.
John Wiley & Sons, Inc., Hoboken, NJ, 2009. xii+393 pp. $110.00. ISBN 9780470409336
This book presents modern developments on the following two subjects: understanding the prop
erties of level sets of a given random field X = (Xt, t T) and analysis and computation of the
distribution function of the random variable MT = suptT X(t), provided that X is realvalued.
Chapter 1 of the book contains a number of fundamental classical results on stochastic processes,
for example, Kolmogorov's consistency theorem and the 01 law for Gaussian processes, but
a particular emphasis is placed on sufficient conditions for continuity, H¨older continuity and
differentiability of trajectories of stochastic processes. Most of the results on path regularity are
not restricted to the Gaussian case, and many apply to the multiparameter (i.e. random field)
setting. The last section of this chapter contains Bulinskaya's sufficient condition for a one
parameter process not to have almost surely critical points in a given level set, plus an extension
of Ylvisaker's theorem in the Gaussian case. Specifically, it is shown here that when the mean
of the Gaussian process is bounded from below and its variance is bounded away from zero, the
