General linear-fractional branching processes with discrete time

@article{Lindo2015GeneralLB,
  title={General linear-fractional branching processes with discrete time},
  author={Alexey Lindo and Serik Sagitov},
  journal={Stochastics},
  year={2015},
  volume={90},
  pages={364 - 378}
}
Abstract We study a linear-fractional Bienaymé–Galton–Watson process with a general type space. The corresponding tree contour process is described by an alternating random walk with the downward jumps having a geometric distribution. This leads to the linear-fractional distribution formula for an arbitrary observation time, which allows us to establish transparent limit theorems for the subcritical, critical and supercritical cases. Our results extend recent findings for the linear-fractional… 
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References

SHOWING 1-10 OF 23 REFERENCES
Linear-fractional branching processes with countably many types
General branching processes in discrete time as random trees
The simple Galton-Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random…
The asymptotic composition of supercritical, multi-type branching populations
The life, past and future are described of a typical individual in an old, non-extinct branching population, where individuals may give birth as a point process and have types in an abstract type…
Some limit theorems for positive recurrent branching Markov chains: I
In this paper we consider a Galton-Watson process whose particles move according to a Markov chain with discrete state space. The Markov chain is assumed to be positive recurrent. We prove a law of…
THE CONTOUR OF SPLITTING TREES IS A LÉVY PROCESS
Splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting…
General irreducible Markov chains and non-negative operators: List of symbols and notation
General branching processes as Markov fields
On the Markov renewal theorem
A Note on the Theory of Moment Generating Functions
in which the integral is assumed to converge for a in some neighborhood of the origin, is called the moment generating function of X. In dealing with certain distribution problems, this function has…
An Introduction To Probability Theory And Its Applications
TLDR
A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
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