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1 results

  • Random environment branching processes with equal environmental extinction probabilities

  • Donald C. Raffety
  • Journal:
    Journal of Applied Probability / Volume 10 / Issue 3 / September 1973
    Published online by Cambridge University Press:
    14 July 2016, pp. 659-665
    Print publication:
    September 1973

  • R-positivity theory for Markov chains is used to obtain results for random environment branching processes whose environment random variables are independent and identically distributed and whose environmental extinction probabilities are equal. For certain processes whose eventual extinction is almost sure, it is shown that the distribution of population size conditioned by non-extinction at time n tends to a left eigenvector of the transition matrix. Limiting values of other conditional probabilities are given in terms of this left eigenvector and it is shown that the probability of non-extinction at time n approaches zero geometrically as n approaches ∞. Analogous results are obtained for processes whose extinction is not almost sure.