Central Limit Analogues for Markov Population Processes

1. SUMMARY In this paper the main discussion is concerned with obtaining asymptotic results for sequences of birth and death processes which are similar to the central limit theorem for sequences of univariate random variables. The motivation is the need to obtain useful approximations to the distributions of sample paths of processes which arise as models for population growth, but for which Kolmogorov differential equations are intractable. In the first section, univariate processes are considered, and conditions are given for the weak convergence of Z N (t) = {X N (t) - aN}/N, where {X N (t), N = 1,2,…} is a sequence of ergodic birth and death processes, to those of an Ornstein-Uhlenbeck process N → ∞. A heuristic method is given which may help explain why this convergence holds, and some examples are given for purposes of illustration. The second part deals with multivariate processes, and three examples are considered in detail: a model for the growth of the sexes in a biological population, a multivariate Ehrenfest process, and a model for the growth and interreaction of two cities. The paper concludes with a discussion of various related results. It is shown that in certain special cases it is possible to obtain diffusions other than the Ornstein-Uhlenbeck process as limits. Finally, heavy traffic results are included for congestion situations originally considered in the special case of time-homogenous arrival rates by Kingman. Transient processes such as epidemics are also shown to exhibit a “central limit” behavior.