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

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) = 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 $S(t)$.