Some Aspects of the Emigration-Immigration Process

Abstract

A multivariate emigration-immigration or a Poisson-Markoff process (Bartlett [2], Ruben and Rothschild [9], Patil [8]; cf. also Bartlett [3], p. 78) is a vector stochastic process $\mathbf{n}(t) = (n_1(t), n_2(t), \cdots, n_m(t))$ in continuous time $t$ which is described by the following properties: There exists a complex of non-negative time-independent parameters, $\lambda_{rs}(r, s = 1, 2, \cdots, m, r \neq s), \lambda^\ast_r$ and $\mu_r(r = 1, 2, \cdots, m)$ such that the probability of a change of $\mathbf{n}(t)$ to $(n_1(t), \cdots, n_r(t) - 1, \cdots, n_s(t) + 1, n_{s + 1}(t), \cdots, n_m(t))$ in the time interval $(t, t + h)$ is $\lambda_{rs}n_r(t)h + o(h)$, for $h$ sufficiently small; the probability of a change of $\mathbf{n}(t)$ to $(n_1(t), \cdots, n_r(t) + 1, \cdots, n_s(t) - 1, n_{s + 1}(t), \cdots, n_m(t))$ is $\lambda_{sr}n_s(t)h + o(h)$; the probability of a change of $\mathbf{n}(t)$ to $(n_1(t), \cdots, n_r(t) - 1, \cdots, n_m(t))$ is $\lambda^\ast_rn_r(t)h + o(h)$; the probability of a change of $\mathbf{n}(t)$ to $(n_1(t), \cdots, n_r(t) + 1, \cdots, n_m(t))$ is $\mu_rh + o(h)$; the probability of no change is $1 - \{\sum_r \mu_r + \sum_r (\lambda^\ast_r + \sum_{s \neq r} \lambda_{rs})n_r(t)\}h + o(h).$ Finally, it is assumed that the above probabilities are independent of the past realization of the process. It is readily seen that these assumptions imply that the process is Markovian and strongly stationary. It will be convenient for our purposes to visualize the Poisson-Markoff process more concretely in the following manner: Consider a system consisting of $m + 1$ states $\{E_1, E_2, \cdots, E_m, E^\ast\}$ such that at any point in time an "individual" is in one of these $m + 1$ states. Thus, $\{E_1, \cdots, E_m\}$ may represent $m$ stages of a certain disease while $E^\ast$ represents the healthy state. A population of individuals is being studied such that the number of individuals in $E^\ast$ at any instant of time is effectively infinite. Let $\lambda_{rs}h + o(h)$ be the probability that an individual in state $E_r$ at time $t$ shall be in $E_s$ at time $t + h, \lambda^\ast_rh + o(h)$ the probability that an individual in $E_r$ at time $t$ shall be in $E^\ast$ at time $t + h$, while $\mu_rh + o(h)$ is the probability that precisely one of the (infinitely many) individuals in $E^\ast$ at time $t$ shall be in $E_r$ at time $t + h$; further, assume that the probability of more than one interchange (of an individual from any state to any other state) in $(t, t + h)$ is $o(h)$ and that there is no interaction between the individuals. Let $n_r(t)$ be the number of individuals in $E_r$ at time $t$. Then clearly the various probabilities enumerated in the first paragraph are precisely the probabilities of the various transitions of $\mathbf{n}(t)$ in the infinitesimal interval $(t, t + h)$. In this connexion, it should be noted that the emigration-immigration process is a natural generalization of the time-homogeneous birth and death process, in the sense that when $m = 1$, the former process reduces to a birth and death process, with $\lambda^\ast$ and $\mu$ representing the death and birth parameters. For $m > 1$, the $\lambda$'s and $\mu$'s may be regarded as the instantaneous mean interaction-rates between the states $E_1, \cdots, E_m, E^\ast$. Since, as remarked previously, the process $\{\mathbf{n}(t)\}$ is stationary, the expectation of $\mathbf{n}(t)$ is independent of $t$. Let, then, $\nu \equiv (\nu_1, \cdots, \nu_m) = E\mathbf{n}(t)$. Then the vector $\nu$ together with the interaction parameters ((2.1) and (2.2)) serve to specify the system completely. (Observe that $\nu$ is functionally related to the interaction parameters through (2.2).) From (2.8), $\nu$ may also be regarded as a limiting vector $(t \rightarrow \infty)$ which represents the "steady state" configuration of the system in the specific stochastic equilibrium sense implied by this equation (a configuration is defined by the vector $\mathbf{n}(t)$), while the parameters determine the approach of the system, again in a probabilistic sense, from an initial configuration to the equilibrium configuration. In Section 2, some basic properties of the Poisson-Markoff process are listed, including the mean lifetime and recurrence time of any configuration in both discrete and continuous time. Section 3 contains the main result of this paper, viz., the joint distribution of a finite set of observations of $\mathbf{n}(t)$ in discrete time from the point of view of the joint factorial moment generating function (f.m.g.f.) of this distribution. These results are of intrinsic theoretical interest and will be utilized in a subsequent paper dealing with the estimation of the fundamental interaction parameter in a singly-infinite class of emigration-immigration processes.