In this monograph the authors give a systematic approach to the
probabilistic properties of the fixed point equation X=AX+B. A
probabilistic study of the stochastic recurrence equation
X_t=A_tX_{t-1}+B_t for real- and matrix-valued random variables A_t,
where (A_t, B_t) constitute an iid sequence, is provided. The classical
theory for these equations, including the existence and uniqueness of a
stationary solution, the tail behavior with special emphasis on power
law behavior, moments and support, is presented. The authors collect
recent asymptotic results on extremes, point processes, partial sums
(central limit theory with special emphasis on infinite variance stable
limit theory), large deviations, in the univariate and multivariate
cases, and they further touch on the related topics of smoothing
transforms, regularly varying sequences and random iterative systems.
The text gives an introduction to the Kesten-Goldie theory for
stochastic recurrence equations of the type X_t=A_tX_{t-1}+B_t. It
provides the classical results of Kesten, Goldie, Guivarc'h, and others,
and gives an overview of recent results on the topic. It presents the
state-of-the-art results in the field of affine stochastic recurrence
equations and shows relations with non-affine recursions and
multivariate regular variation.