Standard

Non-standard limits for a family of autoregressive stochastic sequences. / Foss, Sergey; Schulte, Matthias.

In: Stochastic Processes and their Applications, Vol. 142, 12.2021, p. 432-461.

Research output: Contribution to journal › Article › peer-review

Harvard

Foss, S & Schulte, M 2021, 'Non-standard limits for a family of autoregressive stochastic sequences', Stochastic Processes and their Applications, vol. 142, pp. 432-461. https://doi.org/10.1016/j.spa.2021.09.006

APA

Foss, S., & Schulte, M. (2021). Non-standard limits for a family of autoregressive stochastic sequences. Stochastic Processes and their Applications, 142, 432-461. https://doi.org/10.1016/j.spa.2021.09.006

Vancouver

Foss S, Schulte M. Non-standard limits for a family of autoregressive stochastic sequences. Stochastic Processes and their Applications. 2021 Dec;142:432-461. doi: 10.1016/j.spa.2021.09.006

Author

Foss, Sergey ; Schulte, Matthias. / Non-standard limits for a family of autoregressive stochastic sequences. In: Stochastic Processes and their Applications. 2021 ; Vol. 142. pp. 432-461.

BibTeX

@article{bf444e7cc63d44ed8c7c34225060ad62,
title = "Non-standard limits for a family of autoregressive stochastic sequences",
abstract = "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.",
keywords = "Autoregressive model, Characteristic function, Limiting distribution, Normal distribution, Restart mechanism, Stationary distribution",
author = "Sergey Foss and Matthias Schulte",
note = "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. Publisher Copyright: {\textcopyright} 2021 Elsevier B.V.",
year = "2021",
month = dec,
doi = "10.1016/j.spa.2021.09.006",
language = "English",
volume = "142",
pages = "432--461",
journal = "Stochastic Processes and their Applications",
issn = "0304-4149",
publisher = "Elsevier Science Publishing Company, Inc.",

}

RIS

TY - JOUR

T1 - Non-standard limits for a family of autoregressive stochastic sequences

AU - Foss, Sergey

AU - Schulte, Matthias

N1 - 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. Publisher Copyright: © 2021 Elsevier B.V.

PY - 2021/12

Y1 - 2021/12

N2 - 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.

AB - 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.

KW - Autoregressive model

KW - Characteristic function

KW - Limiting distribution

KW - Normal distribution

KW - Restart mechanism

KW - Stationary distribution

UR - http://www.scopus.com/inward/record.url?scp=85115977331&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2021.09.006

DO - 10.1016/j.spa.2021.09.006

M3 - Article

AN - SCOPUS:85115977331

VL - 142

SP - 432

EP - 461

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

ER -

ID: 34347544