## Bernoulli

• Bernoulli
• Volume 6, Number 2 (2000), 323-338.

### Change of measures for Markov chains and the LlogL theorem for branching processes

Krishna B. Athreya

#### 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 Px and$\widetilde P_x$ denote respectively the probability distribution of a Markov chain$\{X_j\}^{\infty}_0$ with X0=x and transition functions P and$\widetilde P$ . Conditions for$\widetilde P_x$ to be dominated by Px or to be singular with respect to Px 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.

#### Article information

Source
Bernoulli, Volume 6, Number 2 (2000), 323-338.

Dates
First available in Project Euclid: 12 April 2004

https://projecteuclid.org/euclid.bj/1081788031

Mathematical Reviews number (MathSciNet)
MR2001g:60202

Zentralblatt MATH identifier
0969.60076

#### Citation

Athreya, Krishna B. Change of measures for Markov chains and the LlogL theorem for branching processes. Bernoulli 6 (2000), no. 2, 323--338. https://projecteuclid.org/euclid.bj/1081788031

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