Bernoulli

  • Bernoulli
  • Volume 5, Number 6 (1999), 969-998.

Revisiting conditional limit theorems for the mortal simple branching process

Anthony G. Pakes

Full-text: Open access

Abstract

The various conditional limit theorems for the simple branching process are considered within a unified setting. In the subcritical case conditioning events have the form { cal Hn+cal S} , where cal H is the time to extinction, and cal S is a subset of the natural numbers. The resulting limit theorems contain all known forms, and collectively they are equivalent to the classical Yaglom form. In the critical case discrete limits exist provided cal S is a finite set. The principal results are extended to absorbing Markov chains. The Yaglom and Harris theorems for the critical case are generalized by considering the joint behaviour of generation sizes and total progeny conditioned by one-parameter families of events of the form { n<cal Hαn} and { cal H>αn} , where 1 α . A simple representation of the marginal limit laws of the population sizes relates the Yaglom and Harris limits. Analogous structure is elucidated for the marginal limit laws of the total progeny.

Article information

Source
Bernoulli, Volume 5, Number 6 (1999), 969-998.

Dates
First available in Project Euclid: 23 March 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1143122298

Mathematical Reviews number (MathSciNet)
MR1735780

Zentralblatt MATH identifier
0956.60098

Keywords
branching process conditional limit theorem diffusion approximation extinction time total progeny

Citation

Pakes, Anthony G. Revisiting conditional limit theorems for the mortal simple branching process. Bernoulli 5 (1999), no. 6, 969--998. https://projecteuclid.org/euclid.bj/1143122298


Export citation