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Large Deviation Lower Bounds for Additive Functionals of Markov Processes

Naresh C. Jain
The Annals of Probability
Vol. 18, No. 3 (Jul., 1990), pp. 1071-1098
Stable URL: http://www.jstor.org/stable/2244416
Page Count: 28
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Large Deviation Lower Bounds for Additive Functionals of Markov Processes
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Abstract

Let X1, X2,... be a Markov process with state space E, a Polish space. Let Ln(ω, A) = n-1∑n - 1 j = 01A(Xj(ω)) denote the normalized occupation time measure. If μ is a probability measure on E, G is a weak neighborhood of μ, and if $V \subset E$, then we obtain asymptotic lower bounds for probabilities Px[ Ln(ω, ·) ∈ G, Xj(ω) ∈ V, 0 ≤ j ≤ n - 1 ] in terms of I(μ), the rate function of Donsker and Varadhan. Our assumptions are weaker than those imposed by Donsker and Varadhan, and the proof works without any essential change in the continuous time case as well. In fact, the same proofs apply to certain bounded additive functionals: Let r ≥ 0 and let f: Ω → B be bounded F0 r-measurable, where Ω is the sample space with the product topology (Skorohod topology in the continuous time case) and B is a separable Banach space; let θk: Ω → Ω be the shift operator, i.e., θkω(j) = ω(k + j). Then we get lower bounds for probabilities involving n-1(f(ω) + f(θ1ω) + ⋯ + f(θn - 1ω)) in place of Ln(ω, ·). In this latter situation, the rate function has to be the entropy function H(Q) of Donsker and Varadhan.

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