Abstract
We consider the large deviations properties of the empirical measure for one dimensional constrained processes, such as reflecting Brownian motion, the M/M/1 queue, and discrete time analogues. Because these processes do not satisfy the strong stability assumptions that are usually assumed when studying the empirical measure, there is significant probability (from the perspective of large deviations) that the empirical measure charges the point at infinity. We prove the large deviation principle and identify the rate function for the empirical measure for these processes. No assumption of any kind is made with regard to the stability of the underlying process.
Highlights
Let {Xn, n ∈ IN0} be a Markov process on a Polish space S, with transition kernel p(x, dy)
A condition of this sort implies a strong restoring force towards bounded sets, and a force that grows without bound as x → ∞. It is required for a simple reason, and that is to keep the probability that the occupation measure “charges” points at ∞ small from a large deviation perspective
This paper considers a much more general form of the function F, and we prove the full Laplace principle for the empirical measures {Ln, n ∈ IN } when considered as elements of P(IR +), where P(IR +) is the space of probability measures on the one point compactification of IR+
Summary
Let {Xn, n ∈ IN0} be a Markov process on a Polish space S, with transition kernel p(x, dy). A condition of this sort implies a strong restoring force towards bounded sets, and a force that grows without bound as x → ∞ It is required for a simple reason, and that is to keep the probability that the occupation measure “charges” points at ∞ small from a large deviation perspective. In the present paper we consider a class of one dimensional reflected (or constrained) processes which includes the last two examples, and obtain the large deviation principle for the empirical measures without any stability assumptions at all. We end the paper with an Appendix which gives details for some of the proofs that are either standard or similar to others in the paper, and a list of notation is collected there for the reader’s convenience
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