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Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes

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1.	Title	Title of document	Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes
2.	Creator	Author's name, affiliation, country	Ioannis Kontoyiannis; Brown University
2.	Creator	Author's name, affiliation, country	Sean P Meyn; University of Illinois, Urbana-Champaign
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Markov process, large deviations, entropy, Lyapunov function, empirical measures, nonlinear generator, large deviations principle
4.	Description	Abstract	In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn (2003). We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process $\{\Phi(t)\}$ with transition kernel $P$ on a general state space $X$, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals $F$ on $X$, the kernel $\hat P(x,dy) = e^{F(x)} P(x,dy)$ has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a "maximal," well-behaved solution to the "multiplicative Poisson equation," defined as an eigenvalue problem for $\hat P$. Multiplicative Mean Ergodic Theorem: Consider the partial sums of this process with respect to any one of the functionals $F$ considered above. The normalized mean of their moment generating function (and not the logarithm of the mean) converges to the above maximal eigenfunction exponentially fast. Multiplicative regularity: The Lyapunov drift criterion under which our results are derived is equivalent to the existence of regeneration times with finite exponential moments for the above partial sums. Large Deviations: The sequence of empirical measures of the process satisfies a large deviations principle in a topology finer that the usual tau-topology, generated by the above class of functionals. The rate function of this LDP is the convex dual of logarithm of the above maximal eigenvalue, and it is shown to coincide with the Donsker-Varadhan rate function in terms of relative entropy. Exact Large Deviations Asymptotics: The above partial sums are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	NSF, Sloan Foundation
7.	Date	(YYYY-MM-DD)	2005-02-24
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/231
10.	Identifier	Digital Object Identifier	10.1214/EJP.v10-231
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 10
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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