Large Deviations for an Escaping Measure

Details Large Deviations for an Escaping Measure Large Deviations for an Escaping Measure

2012-03-18

arxiv.org/abs/1203.4004v2

Mathematics

Probability

Scientific paper

In this paper, the large deviations on trajectory level is studied for a Markov ergodic process $\xi$. The process $\xi$ is defined on $\R_+$ with values in $\Z^2_+$. A typical path of $\xi$ is a sequence of of excursions in the positive part of $\Z^2_+$ between visits of boundary $\partial\Z^2_+=\{(z_1,0)\in\Z^2_+\}\cup\{(0,z_2)\in\Z^2_+\}$. The scaled processes $\xi_T(\cdot)=\frac{\xi(\cdot T)}{T}$ converge to zero as $T\to\infty$. A type of the large deviation principle with rate function I(\un{f})=\int_0^1(c_1f_1(t)+c_2f_2(t)+c_3\min\{f_1(t),f_2(t)\})dt is proved. Here $\un{f}=(f_1,f_2)\in\R_+^2$ is a continuous function, and constants $c-1, c_2, c_3$ are parameters of the process $\xi$. The local large deviation principle is \lim_{\varepsilon\to 0}\lim_n \lim_{T\to \infty}\frac{1}{T^2}\ln {\bf P}(\xi_T\in U_{n,\varepsilon}(\un{f})=-I(\un{f}), where $U_{n,\varepsilon}$ is a neighborhood of $\un{f}$. The large deviation principle is proved as well, however in a restricted form.

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