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August, 1971 Some Asymptotic Results in a Model of Population Growth II. Positive Recurrent Chains
Burton Singer
Ann. Math. Statist. 42(4): 1296-1315 (August, 1971). DOI: 10.1214/aoms/1177693242


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 $P_{ik}(t), i, k = 0,1,2,\cdots, t \geqq 0$. 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 $P_{ik}(t)$ with $k = k(t) \rightarrow \infty, t \rightarrow \infty$ are required to establish asymptotic formulas for moments of $S(t)$.


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Burton Singer. "Some Asymptotic Results in a Model of Population Growth II. Positive Recurrent Chains." Ann. Math. Statist. 42 (4) 1296 - 1315, August, 1971.


Published: August, 1971
First available in Project Euclid: 27 April 2007

zbMATH: 0303.60080
MathSciNet: MR307372
Digital Object Identifier: 10.1214/aoms/1177693242

Rights: Copyright © 1971 Institute of Mathematical Statistics


Vol.42 • No. 4 • August, 1971
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