 
Summary: Zentralblatt MATH Database 1931 2009
c 2009 European Mathematical Society, FIZ Karlsruhe & SpringerVerlag
Zbl pre05522304
Aza¨is, JeanMarc; Wschebor, Mario
Level sets and extrema of random processes and fields. (English)
Hoboken, NJ: John Wiley & Sons. xi, 393 p. EUR 91.70; £ 73.50 (2009). ISBN
9780470409336/hbk
Given a stochastic process X = {X(t) : t T} with regular paths indexed by a param
eter set T the authors investigate mainly the two following problems: firstly the study
of the properties of the level sets {t T : X(t) = u} for a given u in the state space
of X, and secondly to compute, when the set T is an Euclidian space, the distribution
function FMT (u) = P(MT u) of the supremum MT = suptT X(t) of the field X.
The main tools used to deal with the former problem are the Rice formulas which allow
one to express the factorial moments of the number of roots of X(t) = u as an integral
of a function on the joint distribution of the process and its derivative when T is a
Borel set in Rd
and X takes its values in Rd
. When the preceeding dimensions differ,
more general geometric arguments have to enter into the analysis (the Chapter 6 is
devoted to the derivation of different Rice formulas whether d = d or not). In Chapter
