The Annals of Mathematical Statistics

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

Burton Singer

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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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Ann. Math. Statist., Volume 42, Number 4 (1971), 1296-1315.

First available in Project Euclid: 27 April 2007

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Singer, Burton. Some Asymptotic Results in a Model of Population Growth II. Positive Recurrent Chains. Ann. Math. Statist. 42 (1971), no. 4, 1296--1315. doi:10.1214/aoms/1177693242.

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See also

  • Part I: Burton Singer. Some Asymptotic Results in a Model of Population Growth. Ann. Math. Statist., Volume 41, Number 1 (1970), 115--132.