Non-standard limits for a family of autoregressive stochastic sequences

Sergey Foss, Matthias Schulte

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)432-461
Number of pages30
JournalStochastic Processes and their Applications
Early online date20 Sept 2021
Publication statusPublished - Dec 2021


  • Autoregressive model
  • Characteristic function
  • Limiting distribution
  • Normal distribution
  • Restart mechanism
  • Stationary distribution

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics


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