The ODE Method and Spectral Theory of Markov Operators
Abstract
We give a development of the ODE method for the analysis of recursive algorithms described by a stochastic recursion. With variability modelled via an underlying Markov process, and under general assumptions, the following results are obtained: 1. Stability of an associated ODE implies that the stochastic recursion is stable in a strong sense when a gain parameter is small. 2. The range of gain-values is quantified through a spectral analysis of an associated linear operator, providing a non-local theory. 3. A second-order analysis shows precisely how variability leads to sensitivity of the algorithm with respect to the gain parameter. All results are obtained within the natural operator-theoretic framework of geometrically ergodic Markov processes.
Cite
@article{arxiv.math/0209277,
title = {The ODE Method and Spectral Theory of Markov Operators},
author = {J. Huang and I. Kontoyiannis and S. P. Meyn},
journal= {arXiv preprint arXiv:math/0209277},
year = {2007}
}
Comments
19 pages, 4 figures