Abstract
We consider a sequence of discrete time branching processes with generation-dependent immigration, where the offspring mean tends to its critical value 1. Using a martingale approach, we prove functional limit theorems for suitable normalized fluctuations of the process around its mean when the mean number of immigrating individuals tends to infinity. The limiting processes are deterministically time-changed Wiener processes with three different non-linear time change functions, depending on the behavior of the mean and the variance of the number of immigrants. For the normalized sequence of processes we obtain a deterministic approximation. Consequences related to the maxima and the total progeny of the process will be discussed.
| Original language | English |
|---|---|
| Pages (from-to) | 348-373 |
| Number of pages | 26 |
| Journal | Stochastic Models |
| Volume | 25 |
| Issue number | 2 |
| DOIs | |
| State | Published - Apr 2009 |
Bibliographical note
Funding Information:This paper is based on a part of results obtained under research project No. IN080396 funded by King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. My sincere thanks to both the referees and the associate editor for careful readings of the first version of the article and for their valuable suggestions.
Keywords
- Branching process
- Maxima
- Skorokhod space
- Time-dependent immigration
- Total progeny
ASJC Scopus subject areas
- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics