  We present a development of parts of rate-distortion theory and pattern-
matching algorithms for lossy data compression, centered around a lossy version
of the Asymptotic Equipartition Property (AEP). This treatment closely
parallels the corresponding development in lossless compression, a point of
view that was advanced in an important paper of Wyner and Ziv in 1989. In the
lossless case we review how the AEP underlies the analysis of the Lempel-Ziv
algorithm by viewing it as a random code and reducing it to the idealized
Shannon code. This also provides information about the redundancy of the
Lempel-Ziv algorithm and about the asymptotic behavior of several relevant
quantities. In the lossy case we give various versions of the statement of the
generalized AEP and we outline the general methodology of its proof via large
deviations. Its relationship with Barron's generalized AEP is also discussed.
The lossy AEP is applied to: (i) prove strengthened versions of Shannon's
source coding theorem and universal coding theorems; (ii) characterize the
performance of mismatched codebooks; (iii) analyze the performance of pattern-
matching algorithms for lossy compression; (iv) determine the first order
asymptotics of waiting times (with distortion) between stationary processes;
(v) characterize the best achievable rate of weighted codebooks as an optimal
sphere-covering exponent. We then present a refinement to the lossy AEP and use
it to: (i) prove second order coding theorems; (ii) characterize which sources
are easier to compress; (iii) determine the second order asymptotics of waiting
times; (iv) determine the precise asymptotic behavior of longest match-lengths.
Extensions to random fields are also given.
