On invariant measures of stochastic recursions in a critical case

2007/08/01 by Dariusz Buraczewski
Economics, Econometrics and Finance · Mathematics · #Advanced Differential Equations and Dynamical Systems #Autoregressive model #Invariant (physics) #Invariant measure #LTI system theory #Measure (data warehouse) #Queueing theory #Random variable #Stochastic process #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60B15 #msc:60G50 #msc:60J10

paper · pdf · doi:10.1214/105051607000000140

published as Annals of Applied Probability 2007, Vol. 17, No. 4, 1245-1272 · Published in at http://dx.doi.org/10.1214/105051607000000140 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

openalex publication_date 2007/08/01 · arxiv created 2007/10/19 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We consider an autoregressive model on ℝ defined by the recurrence equation Xn=AnXn−1+Bn, where (Bn, An) are i.i.d. random variables valued in ℝ×ℝ+ and 𝔼[log A₁]=0 (critical case). It was proved by Babillot, Bougerol and Elie that there exists a unique invariant Radon measure of the process Xn. The aim of the paper is to investigate its behavior at infinity. We describe also stationary measures of two other stochastic recursions, including one arising in queuing theory.

Citations

• The tail of the stationary distribution of a random coefficient AR(q) model
• Potential theory of random walks on Abelian groups
• Random difference equations and Renewal theory for products of random matrices