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Biometrika 1967 54(1-2):251-261; doi:10.1093/biomet/54.1-2.251
© 1967 by Biometrika Trust
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A multivariate immigration with multiple death process and applications to lunar craters

ALLAN H. MARCUS

Case Institute of Technology and Statistical Laboratory, University of Cambridge

We consider a stochastic model of a population composed of several types of individuals in which the Poissonwise arrival of an individual of ‘large’ type can cause the immediate death or loss of a certain number of individuals of smaller type. The survival of various small-type individuals is assumed independent, though possibly a function of the size of victim and killer. The probability generating function version of the forward Kolmogorov equation is derived, and from this explicit formulae for the first two factorial moments derived in the general case. The joint probability generating function of the numbers of individuals of various sizes is obtained explicitly in the case of two types of individuals by means of a direct probabilistic argument. The limiting distribution of small-type individuals is studied numerically and shown to be well approximated by the negative binomial distribution. This stochastic process is then used to approximate the formation and survival of lunar craters, a stochastic covering process.


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