Convergence of symmetric Markov chains on ℤd

Richard F. Bass; Takashi Kumagai; Toshihiro Uemura

doi:10.1007/s00440-009-0224-8

Convergence of symmetric Markov chains on ℤ^d

Richard F. Bass, Takashi Kumagai, Toshihiro Uemura

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

12 Citations (Scopus)

Abstract

For each n let Y_t⁽ⁿ⁾ be a continuous time symmetric Markov chain with state space n^-1ℤ^d. Conditions in terms of the conductances are given for the convergence of the Y_t⁽ⁿ⁾t to a symmetric Markov process Y_t on ℝ^d. We have weak convergence of {Y_t⁽ⁿ⁾}: t ≤ t₀} for every t₀ and every starting point. The limit process Y has a continuous part and may also have jumps.

Original language	English
Pages (from-to)	107-140
Number of pages	34
Journal	Probability Theory and Related Fields
Volume	148
Issue number	1
DOIs	https://doi.org/10.1007/s00440-009-0224-8
Publication status	Published - 2010
Externally published	Yes

Keywords

Central limit theorem
Dirichlet forms
Elliptic diffusions
Markov chains
Non-local
Symmetric
Weak convergence

ASJC Scopus subject areas

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

Access to Document

10.1007/s00440-009-0224-8

Cite this

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title = "Convergence of symmetric Markov chains on ℤd",

abstract = "For each n let Yt(n) be a continuous time symmetric Markov chain with state space n-1ℤd. Conditions in terms of the conductances are given for the convergence of the Yt(n)t to a symmetric Markov process Yt on ℝd. We have weak convergence of {Yt(n)}: t ≤ t0} for every t0 and every starting point. The limit process Y has a continuous part and may also have jumps.",

keywords = "Central limit theorem, Dirichlet forms, Elliptic diffusions, Markov chains, Non-local, Symmetric, Weak convergence",

author = "Bass, {Richard F.} and Takashi Kumagai and Toshihiro Uemura",

note = "Funding Information: R. F. Bass{\textquoteright}s research was partially supported by NSF grant DMS-0601783. T. Kumagai{\textquoteright}s research was partially supported by the Grant-in-Aid for Scientific Research (B) 20340017 (Japan). T. Uemura{\textquoteright}s research was partially supported by the Grant-in-Aid for Scientific Research (C) 20540130 (Japan).",

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AU - Bass, Richard F.

AU - Kumagai, Takashi

AU - Uemura, Toshihiro

N1 - Funding Information: R. F. Bass’s research was partially supported by NSF grant DMS-0601783. T. Kumagai’s research was partially supported by the Grant-in-Aid for Scientific Research (B) 20340017 (Japan). T. Uemura’s research was partially supported by the Grant-in-Aid for Scientific Research (C) 20540130 (Japan).

PY - 2010

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N2 - For each n let Yt(n) be a continuous time symmetric Markov chain with state space n-1ℤd. Conditions in terms of the conductances are given for the convergence of the Yt(n)t to a symmetric Markov process Yt on ℝd. We have weak convergence of {Yt(n)}: t ≤ t0} for every t0 and every starting point. The limit process Y has a continuous part and may also have jumps.

AB - For each n let Yt(n) be a continuous time symmetric Markov chain with state space n-1ℤd. Conditions in terms of the conductances are given for the convergence of the Yt(n)t to a symmetric Markov process Yt on ℝd. We have weak convergence of {Yt(n)}: t ≤ t0} for every t0 and every starting point. The limit process Y has a continuous part and may also have jumps.

KW - Central limit theorem

KW - Dirichlet forms

KW - Elliptic diffusions

KW - Markov chains

KW - Non-local

KW - Symmetric

KW - Weak convergence

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