  Stationarity is a very general, qualitative assumption, that can be assessed
on the basis of application specifics. It is thus a rather attractive
assumption to base statistical analysis on, especially for problems for which
less general qualitative assumptions, such as independence or finite memory,
clearly fail. However, it has long been considered too general to allow for
statistical inference to be made. One of the reasons for this is that rates of
convergence, even of frequencies to the mean, are not available under this
assumption alone. Recently, it has been shown that, while some natural and
simple problems such as homogeneity, are indeed provably impossible to solve if
one only assumes that the data is stationary (or stationary ergodic), many
others can be solved using rather simple and intuitive algorithms. The latter
problems include clustering and change point estimation. In this volume I
summarize these results. The emphasis is on asymptotic consistency, since this
the strongest property one can obtain assuming stationarity alone. While for
most of the problems for which a solution is found this solution is
algorithmically realizable, the main objective in this area of research, the
objective which is only partially attained, is to understand what is possible
and what is not possible to do for stationary time series. The considered
problems include homogeneity testing, clustering with respect to distribution,
clustering with respect to independence, change-point estimation, identity
testing, and the general question of composite hypotheses testing. For the
latter problem, a topological criterion for the existence of a consistent test
is presented. In addition, several open questions are discussed.
