Direct Product Branching Processes and Related Induced Markoff Chains I. Calculations of Rates of Approach to Homozygosity

  • Samuel Karlin
  • James Mc Gregor


In this paper we introduce a class of finite state Markoff chains of special structure which includes many cases of interest in applications. For these chains we will determine a full set of eigenvalues (Sections 3 and 4) and provide their probabilistic interpretations (Section 5) which in the present context is rather striking. An intrinsic characterization of these processes and some associated limit theorems will be elaborated in a separate publication (see also [4]).


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  1. [1]
    Feller, W.: Diffusion processes in genetics. Proc. 2nd Berkeley Symp. Mathematical Statistics and Probability. University of California Press 1951, p. 227.Google Scholar
  2. [2]
    Fisher, R. A.: The Genetical Theory of Natural Selection. Oxford: University Press 1930.MATHGoogle Scholar
  3. [3]
    Karlin, S., and J. Mcgregor: The Hahn polynomials, formulas and an application. Scripta Math. 26, 33 (1961).MathSciNetMATHGoogle Scholar
  4. [4]
    Karlin, S., Direct product branching processes and related Markoff chains, Proc. Nat. Acad. Sci., 51 (1964) 598.Google Scholar
  5. [5]
    Kimura, M.: Some problems of stochastic processes in genetics. Ann. Math. Stat. 38, 882 (1957).CrossRefGoogle Scholar
  6. [6]
    Stochastic processes and gene frequencies. Cold Spr. Harb. Symp. Quant. Biol. 20, Population Genetics 1955, 33.Google Scholar
  7. [7]
    Moran, P. A. P.: Statistical Processes in Evolution Theory. Oxford: Clarendon Press 1962.Google Scholar
  8. [8]
    Wright, S.: Evolution in Mendelian populations. Genetics, 16, 97 (1931).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1965

Authors and Affiliations

  • Samuel Karlin
    • 1
  • James Mc Gregor
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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