HOROWITZ SEMINAR

PROBABILITY, ERGODIC THEORY and DYNAMICAL SYSTEMS

Michael Lin (BGU)

will speak on

The CLT for Markov chains.

Abstract:

Let P(x,A) be a Markov transition probability on the general state space (S,Σ), with invariant probability m.
Let {X_n} be the canonical Markov chain on the space of trajectories, with initial distribution m, and assume that the chain is ergodic.
For a function f ε L²(S,m), we are interested in the CENTRAL LIMIT THEOREM for the functional f, which means convergence in distribution of
(1/ n ) Σⁿ_k=1 f(X_k)
to a centered normal distribution N(0, σ_f²).
I will present sufficient conditions, in terms of the growth of the L²-norms of Σⁿ _k=1P^kf,
where Pf(x) =∫ f(y)P(x,dy) is the Markov operator associated to the transition probability.
Special attention will be given to reversible chains.
In addition to the stationary case, in which the initial distribution is the P-invariant probability m, we are also interested in the QUENCHED CENTRAL LIMIT THEOREM:
convergence in distribution when the chain is started from a point, for almost every point in the state space.

To suggest a talk (yours, or someone elses), contact Jon. Aaronson: aaro at post dot tau dot ac dot il

Back to: Jon Aaronson's homepage.

Last update on 12/10/08.