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Limit theorems for stochastic growth models. I

Published online by Cambridge University Press:  01 July 2016

Harry Kesten
Harry Kesten*
Affiliation:
Cornell University
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Abstract

We consider d-dimensional stochastic processes which take values in (R+)d. These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some

Here τ: (R+)d → R+, |x| = Σ1d |x(i)| A = {x ∈ (R+)d: |x| = 1} and T: A → A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that limn→∞Znρ−n = wp. This result is a generalization of the main limit theorem for super-critical branching processes; note, however, that in the present situation both p and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.

Keywords

GROWTH MODELSPOPULATION GROWTHEXPONENTIAL GROWTHMALTHUSIAN PARAMETERGROWTH RATEFIXED POINTPOPULATION COMPOSITIONCONVERGENCE OF COMPOSITION OF POPULATIONFISHER-WRIGHT-HALDANE MODEL
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 2 , August 1972 , pp. 193 - 232
Copyright
Copyright © Applied Probability Trust 1972 

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References

[1] Baum, L. E., Petrie, T., Soules, G. and Weiss, N. (1970) A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. 41, 164–171.Google Scholar
[2] Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass.Google Scholar
[3] Blakley, G. R. (1964) The sequence of iterates of a non negative non-linear transformation. Bull. Amer. Math. Soc. 70, 712–715.Google Scholar
[4] Eilenberg, S. and Steenrod, N. (1952) Foundations of Algebraic Topology. Princeton Univ. Press, Princeton, N. J.CrossRefGoogle Scholar
[5] Feller, W. (1968) An Introduction to Probability Theory and its Applications. Vol. I, 3rd ed. John Wiley, New York.Google Scholar
[6] Karlin, S. (1962–63) Lectures on mathematical biology. Stanford University.Google Scholar
[7] Kesten, H. (1970) Quadratic transformations: a model for population growth. Adv. Appl. Prob. 2, 1–82 and 179–228.Google Scholar
[8] Kesten, H. (1971) Some nonlinear stochastic growth models. Bull. Amer. Math. Soc. 77, 492–511.CrossRefGoogle Scholar
[9] Kesten, H. and Stigum, B. P. Balanced growth under uncertainty in decomposable economies. Submitted to Econometrica. Google Scholar
[10] Li, C. C. (1967) Genetic equilibrium under selection. Biometrics 23, 397–484.Google Scholar
[11] Meyer, P. A. (1966) Probability and Potentials. Blaisdell, Waltham, Mass.Google Scholar
[12] Moran, P. A. P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.Google Scholar
[13] Mulholland, H. P. and Smith, C. A. B. (1959) An inequality in genetical theory. Amer. Math. Monthly 66, 673–683.Google Scholar
[14] Ostrowski, A. M. (1966) Solutions of Equations and Systems of Equations. 2nd ed. Academic Press, New York.Google Scholar
[15] Scheuer, P. A. G. and Mandel, S. P. H. (1959) An inequality in population genetics. Heredity 31, 519–524.Google Scholar
[16] Smale, S. (1963) Stable manifolds for differential equations and diffeomorphisms. Ann. Scuola Norm. Sup. Pisa (3) 17, 97–116.Google Scholar
[17] Sternberg, S. (1957) Local contractions and a theorem of Poincaré. Amer. J. Math. 79, 809–824.CrossRefGoogle Scholar