Absolute continuity of measures corresponding to Markov processes with discrete time

A. A. Lodkin

Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 4, pp. 703-707

Résumé

The aim of the paper is to obtain a generalization of a theorem due to Kukutani [1] concerning the equivalence of product measures. We get a necessary and sufficient condition of absolute continuity for any Markov measures in $X=\prod_{n=1}^\infty Y$. Under the assumption that the "$0-1$" law is valid with respect to $\widetilde\mu$ it can be formulated as follows: $\widetilde\mu\prec\mu$ iff $\mu_n\prec\mu_n$ and $\int\rho_n^{1/q}\,d\mu_n\not\to0$ for every $q>1$ where $\rho_n=d\widetilde\mu_n/d\mu_n$, $\mu_n$, $\widetilde\mu_n$ are the $n$-dimensional projections of $\mu$ and $\widetilde\mu$. In particular, measures corresponding to chains with a finite number of states and processes with homogeneous and independent increments are considered.