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Some Asymptotic Results in a Model of Population Growth II. Positive Recurrent Chains

Burton Singer
The Annals of Mathematical Statistics
Vol. 42, No. 4 (Aug., 1971), pp. 1296-1315
Stable URL: http://www.jstor.org/stable/2240030
Page Count: 20
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Some Asymptotic Results in a Model of Population Growth II. Positive Recurrent Chains
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Abstract

We treat a model describing the continued formation and growth of mutant biological populations. At each transition time of a Poisson process a new mutant population begins its evolution with a fixed number of elements and evolves according to the laws of a continuous time positive recurrent Markov Chain Y(t) with stationary transition probabilities Pik(t), i, k = 0,1,2,⋯, t ≥ 0. Our principal concern is the asymptotic behavior of moments and of the distribution function of the functional S(t) = {number of different sizes of mutant populations at time t}. When the recurrence time distribution to any state of the Markov Chain Y(t) has a finite second moment, the moments of S(t) and limit behavior of its distribution function are controlled by the stationary measure associated with Y(t). When the second moment of the recurrence time distribution is infinite, then a local limit theorem and speed of convergence estimate for Pik(t) with k = k(t) → ∞, t → ∞ are required to establish asymptotic formulas for moments of S(t).

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