On the emergence of random initial conditions in fluid limits

A. D. Barbour*, P. Chigansky, F. C. Klebaner

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


In the paper we present a phenomenon occurring in population processes that start near 0 and have large carrying capacity. By the classical result of Kurtz (1970), such processes, normalized by the carrying capacity, converge on finite intervals to the solutions of ordinary differential equations, also known as the fluid limit. When the initial population is small relative to the carrying capacity, this limit is trivial. Here we show that, viewed at suitably chosen times increasing to ∞, the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition. This random initial condition is related to the martingale limit of an associated linear birth-and-death process.

Original languageAmerican English
Pages (from-to)1193-1205
Number of pages13
JournalJournal of Applied Probability
Issue number4
StatePublished - 1 Dec 2016

Bibliographical note

Publisher Copyright:
© Applied Probability Trust 2016.


  • Birth-death process
  • fluid approximation
  • population dynamics with carrying capacity


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