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Iterated random functions and slowly varying tails

posted 9 Sep 2015, 13:55 by Piotr Dyszewski   [ updated 15 Jun 2017, 01:52 ]
Paper published in Stochastic Processes and their Applications (final versionarXiv)

Abstract:
Consider a sequence of i.i.d. random Lipschitz functions {Ψn}n≥0. Using this sequence we can define a Markov chain via the recursive formula Rn+1=Ψn+1(Rn). It is a well known fact that under some mild moment assumptions this Markov chain has a unique stationary distribution. We are interested in the tail behaviour of this distribution in the case when Ψ(t)≈At+B. We will show that under subexponential assumptions on the random variable log+(A∨B) the tail asymptotic in question can be described using the integrated tail function of log+(A0∨B0). In particular we will obtain new results for the random difference equation Rn+1=An+1Rn+Bn+1.

Talks given on the subject:
  • Iterated Lipschitz maps with slowly varying tails, 44th Probability Summer School, Saint Flour, France, 2014 
  • Perpetuities with slowly varying tails, Probabilistic Aspect of Harmonic Analysis, Będlewo, Poland, 2014, 
  • Perpetuities with slowly varying tails, seminarium ”Analiza Funkcjonalna”, Wrocław, Poland, 2014