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1 results

  • The fractional linear probability generating function in the random environment branching process

  • Part of
  • D. R. Grey, Lu Zhunwei
  • Journal:
    Journal of Applied Probability / Volume 31 / Issue 1 / March 1994
    Published online by Cambridge University Press:
    14 July 2016, pp. 38-47
    Print publication:
    March 1994

  • In a branching process with random environments, the probability of ultimate extinction is a function of the environment sequence, and is therefore a random variable. Explicit results about the distribution of this random variable are difficult to obtain in general. Here we assume independent and identically distributed environments and use the special properties of fractional linear generating functions to derive some explicit distributions, which may be singular or absolutely continuous, depending on the values of certain parameters. We also consider briefly tail behaviour close to 1, and provide an extension to cases where probability generating functions are not fractional linear.