Cutoffs for product chains

Guan Yu Chen; Takashi Kumagai

doi:10.1016/j.spa.2018.01.002

Cutoffs for product chains

Guan Yu Chen, Takashi Kumagai^*

^*Corresponding author for this work

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

2 Citations (Scopus)

Abstract

We consider products of ergodic Markov chains and discuss their cutoffs in total variation. Our framework is general in that rates to pick up coordinates are not necessary equal, and different coordinates may correspond to distinct chains. We give necessary and sufficient conditions for cutoffs of product chains in terms of those of coordinate chains under certain conditions. A comparison of mixing times between the product chain and its coordinate chains is made in detail as well. Examples are given to show that neither cutoffs for product chains nor for coordinate chains imply others in general.

Original language	English
Pages (from-to)	3840-3879
Number of pages	40
Journal	Stochastic Processes and their Applications
Volume	128
Issue number	11
DOIs	https://doi.org/10.1016/j.spa.2018.01.002
Publication status	Published - 2018 Nov
Externally published	Yes

Keywords

Cutoffs
Product chains
Total variation and Hellinger distances

ASJC Scopus subject areas

Statistics and Probability
Modelling and Simulation
Applied Mathematics

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10.1016/j.spa.2018.01.002

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title = "Cutoffs for product chains",

abstract = "We consider products of ergodic Markov chains and discuss their cutoffs in total variation. Our framework is general in that rates to pick up coordinates are not necessary equal, and different coordinates may correspond to distinct chains. We give necessary and sufficient conditions for cutoffs of product chains in terms of those of coordinate chains under certain conditions. A comparison of mixing times between the product chain and its coordinate chains is made in detail as well. Examples are given to show that neither cutoffs for product chains nor for coordinate chains imply others in general.",

keywords = "Cutoffs, Product chains, Total variation and Hellinger distances",

author = "Chen, {Guan Yu} and Takashi Kumagai",

note = "Funding Information: We thank Laurent Saloff-Coste for recalling us that the Hellinger distance was used in the proof of Kakutani{\textquoteright}s dichotomy theorem. The first author is partially supported by grant MOST 104-2115-M-009-013-MY3 and by National Center for Theoretical Science . The second author is partially supported by the Grant-in-Aid for Scientific Research (A) 25247007 and 17H01093 , Japan. Funding Information: We thank Laurent Saloff-Coste for recalling us that the Hellinger distance was used in the proof of Kakutani's dichotomy theorem. The first author is partially supported by grant MOST104-2115-M-009-013-MY3 and by National Center for Theoretical Science. The second author is partially supported by the Grant-in-Aid for Scientific Research (A)25247007 and 17H01093, Japan. Publisher Copyright: {\textcopyright} 2018 Elsevier B.V.",

year = "2018",

month = nov,

doi = "10.1016/j.spa.2018.01.002",

language = "English",

volume = "128",

pages = "3840--3879",

journal = "Stochastic Processes and their Applications",

issn = "0304-4149",

publisher = "Elsevier",

number = "11",

}

TY - JOUR

T1 - Cutoffs for product chains

AU - Chen, Guan Yu

AU - Kumagai, Takashi

N1 - Funding Information: We thank Laurent Saloff-Coste for recalling us that the Hellinger distance was used in the proof of Kakutani’s dichotomy theorem. The first author is partially supported by grant MOST 104-2115-M-009-013-MY3 and by National Center for Theoretical Science . The second author is partially supported by the Grant-in-Aid for Scientific Research (A) 25247007 and 17H01093 , Japan. Funding Information: We thank Laurent Saloff-Coste for recalling us that the Hellinger distance was used in the proof of Kakutani's dichotomy theorem. The first author is partially supported by grant MOST104-2115-M-009-013-MY3 and by National Center for Theoretical Science. The second author is partially supported by the Grant-in-Aid for Scientific Research (A)25247007 and 17H01093, Japan. Publisher Copyright: © 2018 Elsevier B.V.

PY - 2018/11

Y1 - 2018/11

N2 - We consider products of ergodic Markov chains and discuss their cutoffs in total variation. Our framework is general in that rates to pick up coordinates are not necessary equal, and different coordinates may correspond to distinct chains. We give necessary and sufficient conditions for cutoffs of product chains in terms of those of coordinate chains under certain conditions. A comparison of mixing times between the product chain and its coordinate chains is made in detail as well. Examples are given to show that neither cutoffs for product chains nor for coordinate chains imply others in general.

AB - We consider products of ergodic Markov chains and discuss their cutoffs in total variation. Our framework is general in that rates to pick up coordinates are not necessary equal, and different coordinates may correspond to distinct chains. We give necessary and sufficient conditions for cutoffs of product chains in terms of those of coordinate chains under certain conditions. A comparison of mixing times between the product chain and its coordinate chains is made in detail as well. Examples are given to show that neither cutoffs for product chains nor for coordinate chains imply others in general.

KW - Cutoffs

KW - Product chains

KW - Total variation and Hellinger distances

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U2 - 10.1016/j.spa.2018.01.002

DO - 10.1016/j.spa.2018.01.002

M3 - Article

AN - SCOPUS:85041171814

SN - 0304-4149

VL - 128

SP - 3840

EP - 3879

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 11

ER -

Cutoffs for product chains

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Keywords

ASJC Scopus subject areas

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