Large deviations for the empirical measure of a Markov chain with an application to the multivariate empirical measure. (English) Zbl 0661.60043

Ellis, Richard S.

Ann. Probab. 16, No. 4, 1496-1508 (1988).

Let \({\mathcal X}\) be a Polish space, and \b{X}\(=((X_ n),\{P^ x\},\pi (x,dy))\) be an \({\mathcal X}\)-valued Markov chain. The average visiting time, \(L_ n(A)=n^{-1}\sum^{n-1}_{j=0}\delta_{X_ j}(A)\), generates a Borel empirical measure on \({\mathcal X}\). \b{X} is said to satisfy the large deviation property (LDP), if there is a convex lower semicontinuous function J(E) defined on Borel measures in \({\mathcal X}\) with \(J^{-1}[- \infty,L]\) compact, \(L\geq 0\), such that \[ \limsup n^{-1} \log (\sup_{x}P^ x\{L_ n\in E\}\leq (\geq)-J(E) \] for every closed (open) set of measures E, where \(J(E)=\inf \{J(\mu):\mu \in E\}.\)
The main result of this paper proves LDP under the condition \[ \pi^{\beta}(x,A)\leq M\pi^{\beta}(x',A),\quad x,x'\in {\mathcal X},\quad A\quad Borel. \] It extends an analogous result (corresponding to \(\beta =1)\) due to D. W. Stroock [An Introduction to the theory of large deviations (1984; Zbl 0552.60022)].

Reviewer: J.Szulga

MSC:

60F10	Large deviations
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

Polish space; empirical measure; large deviation property

Citations:

Zbl 0552.60022