Limits of renewal processes and Pitman-Yor distribution

2015/04/15 by Bojan Basrak, Basrak, Bojan
Mathematics · #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60F17 #msc:60G55 #msc:60G70

paper · pdf · doi:10.48550/arxiv.1504.03897

14 pages

arxiv created 2015/04/15 · arxiv updated 2015/04/16

Abstract

We consider a renewal process with regularly varying stationary and weakly dependent steps, and prove that the steps made before a given time t, satisfy an interesting invariance principle. Namely, together with the age of the renewal process at time t, they converge after scaling to the Pitman--Yor distribution. We further discuss how our results extend the classical Dynkin--Lamperti theorem.

Related

• Limit Forms of the Distribution of the Number of Renewals
• Limit Theorems for Markov Renewal Processes
• Results for convergence rates associated with renewal processes
• The remainder in the Renewal Theorem
• Perturbation approach to scaled type Markov renewal processes with infinite mean