2017/03/01 by Nicholas Bhattacharya, Bhattacharya, Nicholas, Mark Perlman +1 · 1 citation
Business, Management and Accounting · Economics, Econometrics and Finance · Mathematics · #60J80 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1703.00337
openalex publication_date 2017/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider time-inhomogeneous branching processes and time-inhomogeneous birth-and-death processes, in which the offspring distribution and birth and death rates (respectively) vary in time. A classical result of branching processes states that in the critical regime, a process conditioned on non-extinction and normalized will converge in distribution to a standard exponential. In a paper of Jagers, time-inhomogeneous branching processes are shown to exhibit this convergence as well. In this paper, the hypotheses of Jagers' result are relaxed, further hypotheses are presented for convergence in moments, and the result is extended to the continuous-time analogue of time-inhomogeneous birth-and-death processes. In particular, the new hypotheses suggest a simple characterization of the critical regime.