Publisher DOI: 10.1016/
Title: Non-standard limits for a family of autoregressive stochastic sequences
Language: English
Authors: Foss, Sergey 
Schulte, Matthias 
Keywords: Autoregressive model; Characteristic function; Limiting distribution; Normal distribution; Restart mechanism; Stationary distribution
Issue Date: 20-Sep-2021
Publisher: Elsevier
Source: Stochastic Processes and their Applications 142: 432-461 (2021-12-01)
Abstract (english): 
We examine the influence of using a restart mechanism on the stationary distributions of a particular class of Markov chains. Namely, we consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We show that the restart mechanism may change significantly the limiting distribution. We obtain a limit theorem with a novel type of limiting distribution, a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case without restart mechanism. The main theorem is accompanied by a number of examples and auxiliary results of their own interest.
ISSN: 0304-4149
Journal: Stochastic processes and their applications 
Institute: Mathematik E-10 
Document Type: Article
More Funding information: The work is supported in part by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation.
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