2021/09/30 by Gerold Alsmeyer, Alsmeyer, Gerold
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2109.15007
openalex publication_date 2021/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Linear fractional Galton-Watson branching processes in i.i.d.~random\nenvironment are, on the quenched level, intimately connected to random\ndifference equations by the evolution of the random parameters of their linear\nfractional marginals. On the other hand, any random difference equation defines\nan autoregressive Markov chain (a random affine recursion) which can be\npositive recurrent, null recurrent and transient and which, as the forward\niterations of an iterated function system, has an a.s.~convergent counterpart\nin the positive recurrent case given by the corresponding backward iterations.\nThe present expository article aims to provide an explicit view at how these\naspects of random difference equations and their stationary limits, called\nperpetuities, enter into the results and the analysis, especially in quenched\nregime. Although most of the results presented here are known, we hope that the\noffered perspective will be welcomed by some readers.\n