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$(M,boldM)$ . Let V(.) be a strictly positive eigenfunction of P with eigenvalue ρ>0. Let $$\tilde{P}(x,dy)\equiv \frac{V\left(y\right)P(x,dy)}{\rho V\left(x\right)}.$$ Then$\tilde{P}(.,.)$ is also a transition function. Let P_x and${\tilde{P}}_x$ denote respectively the probability distribution of a Markov chain$\{X_j{\}}_0^{\mathrm{\infty }}$ with X₀=x and transition functions P and$\tilde{P}$ . Conditions for${\tilde{P}}_x$ to be dominated by P_x or to be singular with respect to P_x are given in terms of the martingale sequence$W_n\equiv V\left(X_n\right)/{\rho }^n$ and its limit. This is applied to establish an LlogL theorem for supercritical branching processes with an arbitrary type space.