Deterministic Approximation of a Sequence of Nearly Critical Branching Processes

Abstract

We consider a sequence of discrete time nearly critical branching processes with time-dependent immigration. Using a martingale approach, we prove that when the immigration mean tends to infinity depending on the time of immigration, the suitable normalized sequence can be approximated in Skorokhod metric by a deterministic process. Consequences related to the maxima and the total progeny of the process will be discussed.

Keywords:

• Branching process
• Maxima
• Skorokhod space
• Time-dependent immigration
• Total progeny

Mathematics Subject Classification:

• Primary 60J80
• Secondary 62F12, 60G99

This article is based on a part of results obtained under research project No. IN080396 funded by KFUPM, Dhahran, Saudi Arabia. My sincere thanks to King Fahd University of Petroleum and Minerals for support and facilities.