This book studies the foundations of the theory of linear and nonlinear
forms in single and multiple random variables including the single and
multiple random series and stochastic integrals, both Gaussian and
non-Gaussian. This subject is intimately connected with a number of
classical problems of probability theory such as the summation of
independent random variables, martingale theory, and Wiener's theory of
polynomial chaos. The book contains a number of older results as well as
more recent, or previously unpublished, results. The emphasis is on
domination principles for comparison of different sequences of random
variables and on decoupling techniques. These tools prove very useful in
many areas ofprobability and analysis, and the book contains only their
selected applications. On the other hand, the use of the Fourier
transform - another classical, but limiting, tool in probability
theory - has been practically eliminated. The book is addressed to
researchers and graduate students in prob- ability theory, stochastic
processes and theoretical statistics, as well as in several areas
oftheoretical physics and engineering. Although the ex- position is
conducted - as much as is possible - for random variables with values in
general Banach spaces, we strive to avoid methods that would depend on
the intricate geometric properties of normed spaces. As a result, it is
possible to read the book in its entirety assuming that all the Banach
spaces are simply finite dimensional Euclidean spaces.
