dr Piotr Dyszewski

 

Abstract:

For a given random sequence (C,T1,T2,…) with nonzero C and a.s. finite number of nonzero Tk, the nonhomogeneous smoothing transform S maps the law of a real random variable X to the law of ∑k≥1TkXk+C, where X1,X2,… are independent copies of X and also independent of (C,T1,T2,…). This law is a fixed point of S if the stochastic fixed-point equation (SFPE) X=d∑k≥1TkXk+C holds true, where =d denotes equality in law. Under suitable conditions including EC=0, S possesses a unique fixed point within the class of centered distributions, called the canonical solution to the above SFPE because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on (C,T1,T2,…) such that the canonical solution exhibits right and/or left Poisson tails and the abscissa of convergence of its moment generating function can be determined. As a particular application, the right tail behavior of the Quicksort distribution is found.

 

Talks given on the subject:

Paper accepted to Stochastic Processes and their Applications (arXiv)

 

 

• 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 

Materiały z odczytu na seminarium zakładowym (www)