Jain, Naresh C. Large deviation lower bounds for additive functionals of Markov processes. (English) Zbl 0713.60037 Ann. Probab. 18, No. 3, 1071-1098 (1990). Let \(X_ 0,X_ 1,X_ 2,..\). be a Markov process taking values in a complete separable metric space E and let \(\Omega\) be the space of doubly infinite sequences (...,\(\omega\) (-1),\(\omega\) (0),\(\omega\) (1),...) equipped with the product topology. By \(\pi\) (t,x,dy) denote the transition probability function of the Markov process and put \[ \psi (x,A)=\sum^{\infty}_{j=1}2^{-j}\pi (j,x,A) \] for \(A\in {\mathcal E}\), the Borel \(\sigma\)-field in E. Assume that \(\pi\) is \(\alpha\)-irreducible, i.e. H(1): there exists a probability measure \(\alpha\) on \({\mathcal E}\) such that \(\alpha (A)>0\Rightarrow\psi (x,A)>0\) for all \(x\in E\). Next, introduce the rate function I of Donsker and Varadhan: given any probability measure \(\mu\) on (E,\({\mathcal E})\), \[ I(\mu)=\sup_{u\in {\mathcal U}}\int_{E}\log \frac{u(x)}{\pi u(x)}d\mu (x), \] where \(\pi u(x)=\int_{E}u(y)\pi (1,x,dy)\) and \({\mathcal U}=\{u:\) \(E\to R\); u continuous and inf u\(>0\}\). In addition to H(1), impose H(2): if \(I(\mu)<\infty\), then \(\mu \ll \beta\), where \(\beta (A)=\int_{E}\psi (x,A)\alpha (dx)\), \(A\in {\mathcal E}\). By \[ L_ n(\omega,A)=n^{- 1}\sum^{n-1}_{s=0}\chi_ A(\omega (s)),\quad A\in {\mathcal E}, \] denote the normalized occupation time measure, \(n\geq 1\). The main Theorem 3.7 of the paper states that if G is a neighbourhood of \(\mu\) in the weak topology, then for every \(x\in E\), \[ (*)\quad \liminf_{n}n^{-1} \log P^ x[L_ n\in G]\geq -I(\mu), \] where \(P^ x\) stands for the distribution of the process starting from x to 0. Under suitable additional assumptions (*) holds uniformly for x in a set \(C\subset E\). To obtain this result the author derives first asymptotic lower bounds for probabilities of the form \[ P^ x[L_ n\in G,\quad L_ n(.,V)=1],\quad V\in {\mathcal E},\quad \mu (V)=1, \] and \[ P^ x[L_ n\in G,\quad L_{\hat n}(.,V)=1,\quad \omega (j)\in A\text{ for some } j,\quad n<j\leq \hat n], \] \(A\in {\mathcal E}\), \(\mu (A)>0\), \(\hat n\sim (1+\epsilon)\cdot n\), \(\epsilon >0\), assuming that \(\mu\) is the marginal of the distribution Q on (\(\Omega\),\({\mathcal F})\) of a stationary (and ergodic) process, \({\mathcal F}=\sigma (\omega (j)\), \(-\infty <j<\infty)\). The method of proofs of the above mentioned large deviation results applies equally well to functionals of the form \[ n^{- 1}\{f(\omega)+f(\theta_ 1\omega)+...+f(\theta_{n-1}\omega)\} \] in place of \(L_ n\), where f is a bounded \(\sigma\) (\(\omega\) (0),...,\(\omega\) (r))-measurable mapping (r\(\geq 0)\) of \(\Omega\) into a separable Banach space B, and \(\theta_ i: \Omega \to \Omega\) is the shift operator \(\theta_ i\omega (s)=\omega (i+s)\). Also the case of a Markov process indexed by the continuous time parameter \(t\in R\) is considered and analogous theorems are proved. Conditions H(1), H(2) (and their analogues in the continuous time case) are weaker than the hypotheses used for the proof of similar results by M. D. Donsker and S. R. S. Varadhan [Commun. Pure Appl. Math. 28, 279-301 (1975; Zbl 0348.60031) and 29, 389-461 (1976; Zbl 0348.60032)] and A. de Acosta [J. Theor. Probab. 3, No.3, 395-431 (1990)]. Reviewer: A.M.Zapala Cited in 1 ReviewCited in 13 Documents MSC: 60F10 Large deviations 60J05 Discrete-time Markov processes on general state spaces 60J25 Continuous-time Markov processes on general state spaces Keywords:large deviation; Markov process; occupation time measure; rate function; weak topology Citations:Zbl 0348.60031; Zbl 0348.60032 PDFBibTeX XMLCite \textit{N. C. Jain}, Ann. Probab. 18, No. 3, 1071--1098 (1990; Zbl 0713.60037) Full Text: DOI