Affiliation:
1. Department of Mathematics and Computer Science , Institute for Mathematical Stochastics , University of Münster , Orléans-Ring 10, 48149 Münster , Germany
Abstract
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.
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Statistics, Probability and Uncertainty,Safety, Risk, Reliability and Quality,Statistics and Probability
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