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Linear Fractional Galton–Watson Processes in Random Environment and Perpetuities

  • Gerold Alsmeyer ORCID logo EMAIL logo
Veröffentlicht/Copyright: 20. Oktober 2021
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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.

MSC 2010: 60J80; 60K37; 60H25

Award Identifier / Grant number: EXC 2044–390685587

Funding statement: Work partially funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.

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Received: 2021-08-26
Revised: 2021-09-30
Accepted: 2021-09-30
Published Online: 2021-10-20
Published in Print: 2022-01-01

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