On Asymptotic Structure of the Critical Galton–Watson Branching Processes with Infinite Variance and Allowing Immigration

2021 ◽  
pp. 185-194
Author(s):  
Azam A. Imomov ◽  
Erkin E. Tukhtaev
Keyword(s):  
Branching Processes ◽  
Infinite Variance ◽  
Asymptotic Structure

On distribution tails and expectations of maxima in critical branching processes

1996 ◽  
Vol 33 (3) ◽  
pp. 614-622 ◽  
Author(s):  
K. A. Borovkov ◽  
V A. Vatutin
Keyword(s):  
Related Problem ◽  
Branching Processes ◽  
Domain Of Attraction ◽  
Infinite Variance ◽  
Global Maximum ◽  
Stable Law ◽  
Distribution Tail ◽  
Limit Behaviour ◽  
Watson Process

We derive the limit behaviour of the distribution tail of the global maximum of a critical Galton–Watson process and also of the expectations of partial maxima of the process, when the offspring law belongs to the domain of attraction of a stable law. Thus the Lindvall (1976) and Athreya (1988) results are extended to the infinite variance case. It is shown that in the general case these two asymptotics are closely related to each other, and the latter follows readily from the former. We also discuss a related problem from the theory of general branching processes.

Regularly varying functions in the theory of simple branching processes

1974 ◽  
Vol 6 (3) ◽  
pp. 408-420 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Invariant Measures ◽  
Function Theory ◽  
Branching Processes ◽  
Regularly Varying Function ◽  
Asymptotic Structure ◽  
Regularly Varying Functions ◽  
Limit Laws ◽  
Intimate Connection ◽  
Similar Fact ◽  
Regularly Varying

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.

scholarly journals On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance

2021 ◽  
Vol 73 (8) ◽  
pp. 1056-1066
Author(s):  
A. Imomov ◽  
A. Meyliyev
Keyword(s):  
Generating Function ◽  
Critical Case ◽  
Branching Processes ◽  
Infinite Variance ◽  
Second Moment ◽  
Basic Lemma

UDC 519.218.2 We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remainder. We improve the basic lemma of the theory of critical Markov branching processes and refine known limit results.

scholarly journals On Estimation of the Convergence Rate to Invariant Measures in Markov Branching Processes with Possibly Infinite Variance and Allowing Immigration

2021 ◽  
Vol 14 (5) ◽  
pp. 573-583
Author(s):  
Azam A. Imomov ◽  
Keyword(s):  
Generating Functions ◽  
Continuous Time ◽  
Branching Process ◽  
Critical Case ◽  
Invariant Measures ◽  
Branching Processes ◽  
Immigration Law ◽  
Infinite Variance ◽  
Transition Functions ◽  
First Moment

The paper discusses the continuous-time Markov Branching Process allowing Immigration. We are considering a critical case for which the second moment of offspring law and the first moment of immigration law are possibly infinite. Assuming that the nonlinear parts of the appropriate generating functions are regularly varying in the sense of Karamata, we prove theorems on convergence of transition functions of the process to invariant measures. We deduce the speed rate of these convergence providing that slowly varying factors are with remainder

Sign in / Sign up

Export Citation Format

Share Document