The Annals of Probability

Representations of Invariant Measures on Multitype Galton-Watson Processes

Fred M. Hoppe

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Abstract

We show that there is a one-to-one correspondence between invariant measures for the noncritical multitype Galton-Watson process and invariant measures for the single type process with a linear offspring probability generating function. Two corollaries emerge as simple applications, the first being Spitzer's Martin boundary representation, the second giving the asymptotic behaviour of the measures. Both require no extra moment assumptions and are valid for the multitype theory.

Article information

Source
Ann. Probab., Volume 5, Number 2 (1977), 291-297.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995854

Digital Object Identifier
doi:10.1214/aop/1176995854

Mathematical Reviews number (MathSciNet)
MR431405

Zentralblatt MATH identifier
0379.60068

JSTOR
links.jstor.org

Subjects
Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]
Secondary: 60F15: Strong theorems

Keywords
Multitype Galton-Watson process invariant measures Abel's equation Schroder's equation Martin boundary regular variation conditional Yaglom limit

Citation

Hoppe, Fred M. Representations of Invariant Measures on Multitype Galton-Watson Processes. Ann. Probab. 5 (1977), no. 2, 291--297. doi:10.1214/aop/1176995854. https://projecteuclid.org/euclid.aop/1176995854


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