THE CONSISTENCY AND ADEQUACY OF THE POISSON-MARKOFF MODEL FOR DENSITY FLUCTUATIONS

Abstract

SUMMARY The Poisson-Markoff or emigration-immigration process (as defined in §1.2) has been used as an approximate model for the number of independently moving particles present in particular regions of space, for example, as an approximate model in spermatozoa studies. The postulation of such a model, valid for all modes of division of the space, is shown to be inconsistent in the sense of leading to contradictions, except when τ, the total interval of time considered, is small; the assumptions (cf. §1.3) about the infinitesimal transition probabilities then ensure the consistency of the model to O (τ). To study the adequacy of the Poisson-Markoff model for some data of spermatozoa counts, the asymptotic χ 2 -goodness of fit test based on serial correlations is employed. The goodness of fit tests indicate a significant departure of the spermatozoa data from the Poisson-Markoff model, the departure being not so striking in the uniregional case as in the multiregional case.