Linear Fractional Galton–Watson Processes in Random Environment and Perpetuities

Alsmeyer Gerold ()
Additional contact information
Alsmeyer Gerold: Department of Mathematics and Computer Science, Institute for Mathematical Stochastics, University of Münster, Orléans-Ring 10, 48149 Münster, Germany

Stochastics and Quality Control, 2021, vol. 36, issue 2, 111-127

Abstract: Linear fractional Galton–Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.

Keywords: Galton–Watson Processes in i.i.d. Random Environment; Linear Fractional Distribution; Iterated Function System; Random Difference Equation; Perpetuity; Extinction Probability; Yaglom-Type Limit Law (search for similar items in EconPapers)
Date: 2021
References: Add references at CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1515/eqc-2021-0037 (text/html)
For access to full text, subscription to the journal or payment for the individual article is required.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bpj:ecqcon:v:36:y:2021:i:2:p:111-127:n:4

Ordering information: This journal article can be ordered from
https://www.degruyter.com/journal/key/eqc/html

DOI: 10.1515/eqc-2021-0037

Access Statistics for this article

Stochastics and Quality Control is currently edited by George P. Yanev

More articles in Stochastics and Quality Control from De Gruyter
Bibliographic data for series maintained by Peter Golla ().