Change of Measures for Markov Chains and the LlogL Theorem for Branching Processes

Abstract

Let P(.,.) be a probability transition function on a measurable space $(M,\\bold M)$ . Let V(.) be a strictly positive eigenfunction of P with eigenvalue ρ>0. Let $$ \\widetilde P(x,\\d y)\\equiv\\frac{V(y)P(x,\\d y)}{\\rho V(x)}. $$ Then $\\widetilde P(.,.)$ is also a transition function. Let P<sub>x</sub> and $\\widetilde P_x$ denote respectively the probability distribution of a Markov chain $\\{X_j\\}^{\\infty}_0$ with X<sub>0</sub>=x and transition functions P and $\\widetilde P$ . Conditions for $\\widetilde P_x$ to be dominated by P<sub>x</sub> or to be singular with respect to P<sub>x</sub> are given in terms of the martingale sequence $W_n\\equiv V(X_n)/\\rho^n$ and its limit. This is applied to establish an LlogL theorem for supercritical branching processes with an arbitrary type space.