Some further results on the birth-and-death process and its integral
- 1 January 1968
- journal article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 64 (1) , 141-154
- https://doi.org/10.1017/s0305004100042651
Abstract
In a simple homogeneous birth-and-death process with λ and μ as the constant birth and death rates respectively, letX(t) denote the population size at timet,Z(t) the number of deaths andN(t) the number of events (births and deaths combined) occurring during (0,t). Also let. The results obtained include the following:(a) An explicit formula for the characteristic quasi-probability generating function of the joint distribution ofX(t),Y(t) andZ(t).(b) LetX(0) = 1. It is shown that, ift→ ∞ while λ ≤ μ,N(t) ↑Na.s., whereNtakes only positive odd integral values. If λ > μ, thenP[N(t) ↑ ∞] = 1 − μ/λ. Given thatN(t)∞, the limiting distribution ofN(t) is similar to that ofN. It was reported earlier (Puri (11)), that the limiting distribution ofY(t) is a weighted average of certain chi-square distributions. It is now found that these weights are nothing but the probabilitiesP[N= 2k+ 1] (k= 0, 1,…).(c) Let λ = μ, andMXω),MYωandMZωbe defined as in (36), then as where the c.f. of (X*;Y*;Z*) is given by (38).(d) Exact expressions for the p.d.f. ofY(t) are derived for the cases (i) λ = 0, μ > 0, (ii) λ > 0, μ = 0. For the case (iii) λ gt; 0, μ > 0, since the complete expression is complicated, only the procedure of derivation is indicated.(e) Finally, it is shown that the regressions ofY(t) and ofZ(t) onX(t) are linear forX(t) ≥ 1.This publication has 9 references indexed in Scilit:
- On the homogeneous birth-and-death process and its integralBiometrika, 1966
- The Theory of Branching ProcessesPublished by Springer Nature ,1963
- Equations for stochastic path integralsMathematical Proceedings of the Cambridge Philosophical Society, 1961
- Strong ratio limit propertyBulletin of the American Mathematical Society, 1961
- On the use of the characteristic functional in the analysis of some stochastic processes occurring in physics and biologyMathematical Proceedings of the Cambridge Philosophical Society, 1951
- Random Fluctuations in the Age-Distribution of a Population Whose Development is Controlled by the Simple “Birth-and-Death” ProcessJournal of the Royal Statistical Society Series B: Statistical Methodology, 1950
- An Artificial Realization of a Simple “Birth-and-Death” ProcessJournal of the Royal Statistical Society Series B: Statistical Methodology, 1950
- Stochastic Processes and Population GrowthJournal of the Royal Statistical Society Series B: Statistical Methodology, 1949
- On the Theory of Age-Dependent Stochastic Branching ProcessesProceedings of the National Academy of Sciences, 1948