Relative frequencies in multitype branching processes

This paper considers the relative frequencies of distinct types of individuals in multitype branching processes. We prove that the frequencies are asymptotically multivariate normal when the initial number of ancestors is large and the time of observation is fixed. The result is valid for any branching process with a finite number of types; the only assumption required is that of independent individual evolutions. The problem under consideration is motivated by applications in the area of cell biology. Specifically, the reported limiting results are of advantage in cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement. Relevant statistical applications are discussed in the context of asymptotic maximum likelihood inference for multitype branching processes.Comment: Published in at http://dx.doi.org/10.1214/08-AAP539 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Two-Type Age-Dependent Branching Processes with Inhomogeneous Immigration as Models of Renewing Cell Population

2000 Mathematics Subject Classification: primary 60J80; secondary 60J85, 92C37.Two-type reducible age-dependent branching processes with inhomogeneous immigration are considered to describe the kinetics of renewing cell populations. This class of processes can be used to model the generation of oligodendrocytes in the central nervous system in vivo or the kinetics of leukemia cells. The asymptotic behavior of the first and second moments, including the correlation, of the process is investigated.This research was supported by NIH grants 2R01 NS039511, R01 CA134839, and 1R01 AI069351

Limiting Distributions for Lifetimes in Alternating Renewal Processes

2000 Mathematics Subject Classification: 60K05The spent life time and the residual life time are well investigated characteristics of an ordinary renewal process. In the present paper a generalization of these lifetime processes associated with an alternating renewal process is considered. Limiting distributions are presented in the case of in finite mean renewal periods.The paper is supported by NFSI-Bulgaria, Grant No. MM-1101/2001

Branching Stochastic Processes: Regulation, Regeneration, Estimation, Applications

2000 Mathematics Subject Classification: 60J80.This is a survey of the works of Bulgarian mathematicians in the area of Branching Stochastic Processes

Continuous-time Controlled Branching Processes

Controlled branching processes with continuous time are introduced and limiting distributions are obtained in the critical case. An extension of this class as regenerative controlled branching processes with continuous time is proposed and some asymptotic properties are considered.KP-6-H22/3 (Bulgarian National Science Fund at the Ministry of Education and Science); the Junta de Extremadura (grant GR18103 and IB16099); the Ministerio de Econom´ıa y Competitividad of Spain (grant MTM2015-70522-P); the Ministerio de Ciencia e Innovaci´on of Spain (grant PID2019-108211GB- I00/AEI/10.13039/501100011033) and the Fondo Europeo de Desarrollo Regional

Dr. Andrei Yakovlev: Visionary, Leader, Iconoclast

http://deepblue.lib.umich.edu/bitstream/2027.42/112886/1/13062_2008_Article_88.pd

Poisson Random Measures and Noncritical Multitype Markov Branching Processes

We investigate noncritical multi-type Markov branching processes with immigration generated by Poisson measures. Limiting distributions are obtained when the rates of the Poisson measures are asymptotically equivalent to exponential or regularly varying functions. In particular, results analogous to a strong LLN are presented, and limiting normal distributions are obtained when the rates increase. When the rates decrease, then conditional limiting distributions are established. A stationary limiting distribution is obtained when the mean Poisson measure grows linearly. The asymptotic behaviour of the first and second moments of the processes is also investigated.NFSR of the Ministry of Education and Science of Bulgaria, grant No. KP–06-H22/3