The ODE Method and Spectral Theory of Markov Operators

Details The ODE Method and Spectral Theory of Markov Operators The ODE Method and Spectral Theory of Markov Operators

2002-09-20

arxiv.org/abs/math/0209277v1

Proceedings of Stochastic Theory and Control Workshop, Springer, New York, pp. 205-221, B. Pasik-Duncan (Editor), 2002

Mathematics

Probability

19 pages, 4 figures

Scientific paper

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.

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