Open AccessOn central limit theorems for branching processes with dependent immigration
Author(s) -
Vitaliy Golomoziy,
S.O. Sharipov
Publication year - 2020
Publication title -
vìsnik. serìâ fìziko-matematičnì nauki/vìsnik kiì̈vsʹkogo nacìonalʹnogo unìversitetu ìmenì tarasa ševčenka. serìâ fìziko-matematičnì nauki
Language(s) - English
Resource type - Journals
eISSN - 2218-2055
pISSN - 1812-5409
DOI - 10.17721/1812-5409.2020/1-2.1
Subject(s) - central limit theorem , infinity , mathematics , branching process , immigration , limit (mathematics) , mathematical proof , branching (polymer chemistry) , variance (accounting) , statistical physics , combinatorics , mathematical analysis , physics , statistics , geography , economics , geometry , accounting , archaeology , materials science , composite material
In this paper we consider subcritical and supercritical discrete time branching processes with generation dependent immigration. We prove central limit theorems for fluctuation of branching processes with immigration when the mean of immigrating individuals tends to infinity with the generation number and immigration process is m−dependent. The first result states on weak convergence of the fluctuation subcritical branching processes with m−dependent immigration to standard normal distribution. In this case, we do not assume that the mean and variance of immigration process are regularly varying at infinity. In contrast, in Theorem 3.2, we suppose that the mean and variance are to be regularly varying at infinity. The proofs are based on direct analytic method of probability theory.