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

Let P(.,.) be a probability transition function on a measurable space (M, M). Let V(.) be a strictly positive eigenfunction of P with eigenvalue ρ > 0. Let$\tilde{P}(x,{\rm d}y)\equiv {\textstyle\frac{V(y)P(x,{\rm d}y)}{\rho V(x)}}$. Then P̃(.,.) is also a transition function. Let Pxand P̃xdenote...

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Bibliographic Details
Published in:Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability 2000-04, Vol.6 (2), p.323-338
Main Author: Athreya, Krishna B.
Format: Article
Language:English
Subjects:
change of measures
Markov chains
Martingales
Mathematical functions
Mathematical theorems
Mathematical vectors
Mathematics
measure-valued branching processes
Perceptron convergence procedure
Probabilities
Probability distributions
Statistical theories
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Summary:Let P(.,.) be a probability transition function on a measurable space (M, M). Let V(.) be a strictly positive eigenfunction of P with eigenvalue ρ > 0. Let$\tilde{P}(x,{\rm d}y)\equiv {\textstyle\frac{V(y)P(x,{\rm d}y)}{\rho V(x)}}$. Then P̃(.,.) is also a transition function. Let Pxand P̃xdenote respectively the probability distribution of a Markov chain {Xj}0 ∞with X0=x and transition functions P and P̃. Conditions for P̃xto be dominated by Pxor to be singular with respect to Pxare 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.
ISSN:1350-7265
DOI:10.2307/3318579