Limit theorems and structural properties of the cat-and-mouse markov chain and its generalisations

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Limit theorems and structural properties of the cat-and-mouse markov chain and its generalisations. / Foss, Sergey; Prasolov, Timofei; Shneer, Seva.

In: Advances in Applied Probability, Vol. 54, No. 1, 28.03.2022, p. 141-166.

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

BibTeX

@article{96b54471ce4c41a9b27aac0fed6838a6,

title = "Limit theorems and structural properties of the cat-and-mouse markov chain and its generalisations",

abstract = "We revisit the so-called cat-and-mouse Markov chain, studied earlier by Litvak and Robert (2012). This is a two-dimensional Markov chain on the lattice <![CDATA[ $\mathbb{Z}^2$ ]]>, where the first component (the cat) is a simple random walk and the second component (the mouse) changes when the components meet. We obtain new results for two generalisations of the model. First, in the two-dimensional case we consider far more general jump distributions for the components and obtain a scaling limit for the second component. When we let the first component be a simple random walk again, we further generalise the jump distribution of the second component. Secondly, we consider chains of three and more dimensions, where we investigate structural properties of the model and find a limiting law for the last component. ",

keywords = "Cat-and-mouse games, compound renewal process, multidimensional Markov chain, randomly stopped sums, regular variation, weak convergence",

author = "Sergey Foss and Timofei Prasolov and Seva Shneer",

note = "Funding Information: Research was supported by the Mathematical Center in Akademgorodok under Agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation. Publisher Copyright: {\textcopyright} The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust.",

year = "2022",

month = mar,

day = "28",

doi = "10.1017/apr.2021.23",

language = "English",

volume = "54",

pages = "141--166",

journal = "Advances in Applied Probability",

issn = "0001-8678",

publisher = "Applied Probability Trust",

number = "1",

}

RIS

TY - JOUR

T1 - Limit theorems and structural properties of the cat-and-mouse markov chain and its generalisations

AU - Foss, Sergey

AU - Prasolov, Timofei

AU - Shneer, Seva

N1 - Funding Information: Research was supported by the Mathematical Center in Akademgorodok under Agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation. Publisher Copyright: © The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust.

PY - 2022/3/28

Y1 - 2022/3/28

N2 - We revisit the so-called cat-and-mouse Markov chain, studied earlier by Litvak and Robert (2012). This is a two-dimensional Markov chain on the lattice <![CDATA[ $\mathbb{Z}^2$ ]]>, where the first component (the cat) is a simple random walk and the second component (the mouse) changes when the components meet. We obtain new results for two generalisations of the model. First, in the two-dimensional case we consider far more general jump distributions for the components and obtain a scaling limit for the second component. When we let the first component be a simple random walk again, we further generalise the jump distribution of the second component. Secondly, we consider chains of three and more dimensions, where we investigate structural properties of the model and find a limiting law for the last component.

AB - We revisit the so-called cat-and-mouse Markov chain, studied earlier by Litvak and Robert (2012). This is a two-dimensional Markov chain on the lattice <![CDATA[ $\mathbb{Z}^2$ ]]>, where the first component (the cat) is a simple random walk and the second component (the mouse) changes when the components meet. We obtain new results for two generalisations of the model. First, in the two-dimensional case we consider far more general jump distributions for the components and obtain a scaling limit for the second component. When we let the first component be a simple random walk again, we further generalise the jump distribution of the second component. Secondly, we consider chains of three and more dimensions, where we investigate structural properties of the model and find a limiting law for the last component.

KW - Cat-and-mouse games

KW - compound renewal process

KW - multidimensional Markov chain

KW - randomly stopped sums

KW - regular variation

KW - weak convergence

UR - http://www.scopus.com/inward/record.url?scp=85125854045&partnerID=8YFLogxK

U2 - 10.1017/apr.2021.23

DO - 10.1017/apr.2021.23

M3 - Article

AN - SCOPUS:85125854045

VL - 54

SP - 141

EP - 166

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 1

ER -

ID: 35663968