# 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…

## 5 Citations

A pathwise iterative approach to the extinction of branching processes with countably many types

- Mathematics
- 2016

We consider the extinction events of Galton-Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton-Watson processes with finite but…

A pathwise approach to the extinction of branching processes with countably many types

- MathematicsStochastic Processes and their Applications
- 2019

Extinction in branching processes with countably many types

- Mathematics
- 2018

Multitype branching processes describe the evolution of populations in which individuals give birth independently according to a probability distribution that depends on their type. In this thesis,…

Regenerative multi-type Galton-Watson processes

- Mathematics
- 2017

The general Perron-Frobenius theorem describes the growth of powers of irreducible non-negative kernels. In the special case of kernels with an atom this result can be obtained using a regeneration…

Perron-Frobenius theory for kernels and Crump-Mode-Jagers processes with macro-individuals

- MathematicsJ. Appl. Probab.
- 2020

A new probabilistic interpretation of the general regeneration method in terms of multi-type Galton-Watson processes producing clusters of particles is given, treating clusters as macro-individuals and arriving at a single-type Crump-Mode-Jagers process with a naturally embedded renewal structure.

## References

SHOWING 1-10 OF 23 REFERENCES

General branching processes in discrete time as random trees

- Mathematics
- 2008

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

- Mathematics
- 1996

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

- MathematicsAdvances in Applied Probability
- 1998

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

- Mathematics
- 2007

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

- Mathematics
- 1984

A Note on the Theory of Moment Generating Functions

- Mathematics
- 1942

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

- Mathematics
- 1950

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.