MPS-RR 2000-47
December 2000

Classification of Markov Chains on $mathbb{R}^{k}$

Niels Richard Hansen

Abstract

In this paper it is shown that the behavior of a Markov chain on $mathbb{R}^{k}$ to a large extent is determined by the conditional mean values and the conditional variances. First it is shown that geometric drift (or drift) towards a compact set using the simple and well known drift function $V(x)=1+|x|^{2}$ is completely characterized by these two conditional moments, and also uniform ergodicity can be derived on the basis of these moments. Secondly, a special class of Markov chains, called affine Markov chains, are considered and a new kind of drift function is introduced. Using this drift function we derive another criteria for geometric drift towards a compact set again based on the two conditional moments.

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