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Ergodicity of age structure in populations with Markovian vital rates. II. General states

Published online by Cambridge University Press:  01 July 2016

Joel E. Cohen
Joel E. Cohen*
Affiliation:
The Rockefeller University, New York
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Abstract

The age structure of a large, unisexual, closed population is described here by a vector of the proportions in each age class. Non-negative matrices of age-specific birth and death rates, called Leslie matrices, map the age structure at one point in discrete time into the age structure at the next. If the sequence of Leslie matrices applied to a population is a sample path of an ergodic Markov chain, then: (i) the joint process consisting of the age structure vector and the Leslie matrix which produced that age structure is a Markov chain with explicit transition function; (ii) the joint distribution of age structure and Leslie matrix becomes independent of initial age structure and of the initial distribution of the Leslie matrix after a long time; (iii) when the Markov chain governing the Leslie matrix is homogeneous, the joint distribution in (ii) approaches a limit which may be easily calculated as the solution of a renewal equation. A numerical example will be given in Cohen (1977).

Keywords

AGE STRUCTUREPOPULATION DYNAMICSERGODIC THEOREMS OF DEMOGRAPHYPRODUCTS OF RANDOM MATRICESMULTITYPE PROCESSES IN RANDOM ENVIRONMENTS
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 1 , March 1977 , pp. 18 - 37
Copyright
Copyright © Applied Probability Trust 1977 

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References

Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Billingsley, P. and Tops⊘e, F. (1967) Uniformity in weak convergence. Z. Wahrscheinlichkeitsth. 7, 1–16.Google Scholar
Blumenthal, R. M. and Getoor, R. K. (1968) Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
Brass, W. (1974) Perspectives in population prediction: illustrated by the statistics of England and Wales. J. R. Statist. Soc. A 137, 532–583.Google Scholar
Cohen, J. E. (1976) Ergodicity of age structure in populations with Markovian vital rates. I. Countable states. J. Amer. Statist. Assoc. 71, 335–339.Google Scholar
Cohen, J. E. (1977) Ergodicity of age structure in populations with Markovian vital rates. III. Mean and approximate variance. Adv. Appl. Prob. 9 (3).Google Scholar
Dobrushin, R. L. (1956) Central limit theorem for nonstationary Markov chains. I, II. Theor. Prob. Appl. 1, 65–79, 329–383.Google Scholar
Dunford, N. and Schwartz, J. T. (1958) Linear Operators. I: General Theory. Interscience, New York.Google Scholar
Golubitsky, M., Keeler, E. B. and Rothschild, M. (1975) Convergence of the age-structure: applications of the projective method. Theoret. Pop. Biol. 7, 84–93.Google Scholar
Griffeath, D. (1975) A maximal coupling for Markov chains. Z. Wahrscheinlichkeitsth. 31, 95–106.Google Scholar
Hajnal, J. (1956) The ergodic properties of nonhomogeneous finite Markov chains. Proc. Camb. Phil. Soc. 52, 67–77.Google Scholar
Hajnal, J. (1958) Weak ergodicity in nonhomogeneous Markov chains. Proc. Camb. Phil. Soc. 54, 233–246.Google Scholar
Hajnal, J. (1976) On products of non-negative matrices. Math. Proc. Camb. Phil. Soc. 79, 521–530.CrossRefGoogle Scholar
Kelley, J. L. (1955) General Topology. Van Nostrand, Princeton, N.J.Google Scholar
Kingman, J. F. C. and Taylor, S. J. (1966) Introduction to Measure and Probability. Cambridge University Press.Google Scholar
Loève, M. (1963) Probability Theory, 3rd edn. Van Nostrand, Princeton, N.J. Google Scholar
Pitman, J. W. (1974) Uniform rates of convergence for Markov chain transition probabilities. Z. Wahrscheinlichkeitsth. 29, 193–227.CrossRefGoogle Scholar
Takahashi, Y. (1969) Markov chains with random transition matrices. Kodai Math. Seminar Rep. 21, 426–447.Google Scholar