## The Annals of Mathematical Statistics

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

Burton Singer

#### 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 $P_{ik}(t), i, k = 0,1,2,\cdots, t \geqq 0$. 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 $S(t)$.

#### Article information

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

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177693242

Digital Object Identifier
doi:10.1214/aoms/1177693242

Mathematical Reviews number (MathSciNet)
MR307372

Zentralblatt MATH identifier
0303.60080

JSTOR