**Abstract:**
For strongly ergodic discrete-time Markov chains we discuss the
possible limits as n tends to infinity of probability measures on
the path space of the form
exp(nH(L_{n}))dP/Z_{n}.
L_{n} is the empirical measure (or sojourn measure)
of the process, H is a real-valued function
(possibly attaining minus infinity)
on the space of probability measures on the state space of the chain,
and Z_{n} is the appropriate norming constant.
The class of these transformations also includes conditional
laws given L_{n} belongs to some set.
The possible limit laws are mixtures of Markov chains minimizing
a certain free energy. The method of proof strongly
relies on large deviation techniques.

**Keywords:**
maximum entropy principle,
large deviations,
Markov chains,
variational problem,
weak convergence

**2010
Mathematics Subject Classification:**

- 60F05 Central limit and other weak theorems
- 60F10 Large deviations
- 60J05 Markov processes with discrete parameter

**Reference:**
Stochastic
Processes and their Applications 33 (1989) 1-27.

DOI: 10.1016/0304-4149(89)90063-X

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