  State space models have long played an important role in signal processing.
The Gaussian case can be treated algorithmically using the famous Kalman
filter. Similarly since the 1970s there has been extensive application of
Hidden Markov models in speech recognition with prediction being the most
important goal. The basic theoretical work here, in the case $X$ and $Y$ finite
(small) providing both algorithms and asymptotic analysis for inference is that
of Baum and colleagues. During the last 30-40 years these general models have
proved of great value in applications ranging from genomics to finance.
  Unless the $X,Y$ are jointly Gaussian or $X$ is finite and small the problem
of calculating the distributions discussed and the likelihood exactly are
numerically intractable and if $Y$ is not finite asymptotic analysis becomes
much more difficult. Some new developments have been the construction of
so-called ``particle filters'' (Monte Carlo type) methods for approximate
calculation of these distributions (see Doucet et al. [4]) for instance and
general asymptotic methods for analysis of statistical methods in HMM [2] and
other authors.
  We will discuss these methods and results in the light of exponential mixing
properties of the conditional (posterior) distribution of $(X_1,X_2,...)$ given
$(Y_1,Y_2,...)$ already noted by Baum and Petrie and recent work of the authors
Bickel, Ritov and Ryden, Del Moral and Jacod, Douc and Matias.
