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**Harvard**

Sagitov, S. (2017) *Tail generating functions for extendable branching processes*.

** BibTeX **

@article{

Sagitov2017,

author={Sagitov, Serik},

title={Tail generating functions for extendable branching processes},

journal={Stochastic Processes and Their Applications},

issn={0304-4149},

volume={127},

issue={5},

pages={1649-1675},

abstract={We study branching processes of independently splitting particles in the continuous time setting. If time is calibrated such that particles live on average one unit of time, the corresponding transition rates are fully determined by the generating function f for the offspring number of a single particle. We are interested in the defective case f (1) = 1 - epsilon, where each splitting particle with probability epsilon is able to terminate the whole branching process. A branching process [Z(t)}(t >= 0) will be called extendable if f (q) = q and f (r) = r for some 0 <= q < r < infinity. Specialising on the extendable case we derive an integral equation for F-t (s) = Es-Zt. This equation is expressed in terms of what we call, tail generating functions. With help of this equation, we obtain limit theorems for the time to termination as epsilon -> 0. We find that conditioned on non-extinction, the typical values of the termination time follow an exponential distribution in the nearly subcritical case, and require different scalings depending on whether the reproduction regime is asymptotically critical or supercritical. Using the tail generating function approach we also obtain new refined asymptotic results for the regular branching processes with f (1) = 1.},

year={2017},

keywords={Markov branching process, Branching with killing, Modified linear-fractional distribution, xlogx-condition},

}

** RefWorks **

RT Journal Article

SR Electronic

ID 249615

A1 Sagitov, Serik

T1 Tail generating functions for extendable branching processes

YR 2017

JF Stochastic Processes and Their Applications

SN 0304-4149

VO 127

IS 5

SP 1649

OP 1675

AB We study branching processes of independently splitting particles in the continuous time setting. If time is calibrated such that particles live on average one unit of time, the corresponding transition rates are fully determined by the generating function f for the offspring number of a single particle. We are interested in the defective case f (1) = 1 - epsilon, where each splitting particle with probability epsilon is able to terminate the whole branching process. A branching process [Z(t)}(t >= 0) will be called extendable if f (q) = q and f (r) = r for some 0 <= q < r < infinity. Specialising on the extendable case we derive an integral equation for F-t (s) = Es-Zt. This equation is expressed in terms of what we call, tail generating functions. With help of this equation, we obtain limit theorems for the time to termination as epsilon -> 0. We find that conditioned on non-extinction, the typical values of the termination time follow an exponential distribution in the nearly subcritical case, and require different scalings depending on whether the reproduction regime is asymptotically critical or supercritical. Using the tail generating function approach we also obtain new refined asymptotic results for the regular branching processes with f (1) = 1.

LA eng

DO 10.1016/j.spa.2016.09.004

LK http://dx.doi.org/10.1016/j.spa.2016.09.004

OL 30