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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big JOINT PROBABILITY AND STATISTICS SEMINAR}
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\centerline{4:15 p.m., Tuesday, February 26, 2002}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Ioannis Kontoyiannis}
\centerline{\sl Brown University}
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\centerline{\bf Limit Theorems and Some Spectral Theory for Geometrically Ergodic Markov Chains}
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Precise limit theorems are developed for Markov chains and Markov
processes on general state spaces, under natural and often minimal
assumptions. Assuming only that the chain is geometrically ergodic,
we obtain Edgeworth expansions, a large deviations principle,
and exact large deviations asymptotics for the partial sums of
a real-valued functional of the chain. In particular, we give
the first complete description of these results for Doeblin chains,
as well as for important models in applications like reflected
Brownian motion. In some special cases it also possible to obtain
non-asymptotic bounds with computable constants.

The gist of the present  approach is the development of
a multiplicative ergodic theory of Markov chains, in close analogy
to the standard additive theory. The main technical step is the 
study
of an eigenvalue problem, the "multiplicative Poisson equation."
This is an exponential version of the classical Poisson equation
in Markov process theory. These results are motivated, in part,
by potential applications in stochastic networks, linear systems,
and simulation.

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