What is the maximum level a certain river is likely to reach over the
next 25 years? (Having experienced three times a few feet of water in my
house, I feel a keen personal interest in this question. ) There are
many questions of the same nature: what is the likely magnitude of the
strongest earthquake to occur during the life of a planned building, or
the speed ofthe strongestwind a suspension bridge will have to stand?
All these situations can be modeled inthesamemanner. Thevalue X of the
quantity of interest (be it water t level or speed of wind) at time t is
a random variable. What can be said about the maximum value of X over a
certain range of t? t A collection of random variables (X ), where t
belongs to a certain index t set T, is called a stochastic process, and
the topic of this book is the study of the supremum of certain
stochastic processes, and more precisely to ?nd upper and lower bounds
for the quantity EsupX . (0. 1) t t?T Since T might be uncountable, some
care has to be taken to de?ne this quantity. For any reasonable
de?nition of Esup X we have t t?T EsupX =sup{EsupX; F?T, F ?nite}, (0.
2) t t t?T t?F an equality that we will take as the de?nition of the
quantity Esup X . t t?T Thus, the crucial case for the estimation of the
quantity (0.