Svetlana Bizjajeva and Jimmy Olsson

General State Space Markov Chains

Abstract:

The  theory of  countable  state  space Markov  chains  is a  standard
element of  graduate courses in  probability and is familiar  to most
people in the  field of mathematical statistics. The  aim of the talk
is  to, with  the theory  of countable  spaces as  a  starting point,
present  an  overview  of   the  corresponding  general  state  space
theory. The survey is based on the pioneering work "Markov chains and
stochastic stability"  by Meyn and  Tweedie (1993) and starts  with a
presentation  of  the  basic   definitions  and  properties  such  as
transition    kernels,    the     strong    Markov    property    and
irreducibility. This is followed by a description of how parts of the
theory  of general  spaces can  be transferred  to that  of countable
spaces  through  the concept  of  atoms  and  the Nummelin  splitting
technique. It then passes on  to a discussion of stability structures
such   as   transience/recurrence,   Harris   recurrence,   invariant
distributions, regularity  and drift  conditions, and ends  with some
results on ergodicity, that is the convergence of the distribution of
the chain to its stationary distribution.