Weak Invariance Principle for the Local Times of Partial Sums of Markov Chains

Michael Bromberg; Zemer Kosloff

doi:https://doi.org/10.1007/s10959-012-0438-z

Weak Invariance Principle for the Local Times of Partial Sums of Markov Chains

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

Abstract

Let {X_n} be an integer-valued Markov chain with finite state space. Let S_n = Σ_k=0ⁿ X_k and let L_n(x) be the number of times S_k hits x ∈ ℤ up to step n. Define the normalized local time process l_n(t,x) by (Formula presented.). The subject of this paper is to prove a functional weak invariance principle for the normalized sequence l_n(t,x), i.e., we prove under the assumption of strong aperiodicity of the Markov chain that the normalized local times converge in distribution to the local time of the Brownian motion.

Original language	English
Pages (from-to)	493-517
Number of pages	25
Journal	Journal of Theoretical Probability
Volume	27
Issue number	2
DOIs	https://doi.org/10.1007/s10959-012-0438-z
State	Published - Jun 2014
Externally published	Yes

Keywords

Brownian motion
Local times
Markov chains
Weak invariance principle

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty
General Mathematics

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https://doi.org/10.1007/s10959-012-0438-z

Cite this

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title = "Weak Invariance Principle for the Local Times of Partial Sums of Markov Chains",

abstract = "Let {Xn} be an integer-valued Markov chain with finite state space. Let Sn = Σk=0n Xk and let Ln(x) be the number of times Sk hits x ∈ ℤ up to step n. Define the normalized local time process ln(t,x) by (Formula presented.). The subject of this paper is to prove a functional weak invariance principle for the normalized sequence ln(t,x), i.e., we prove under the assumption of strong aperiodicity of the Markov chain that the normalized local times converge in distribution to the local time of the Brownian motion.",

keywords = "Brownian motion, Local times, Markov chains, Weak invariance principle",

author = "Michael Bromberg and Zemer Kosloff",

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language = "الإنجليزيّة",

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TY - JOUR

T1 - Weak Invariance Principle for the Local Times of Partial Sums of Markov Chains

AU - Bromberg, Michael

AU - Kosloff, Zemer

PY - 2014/6

Y1 - 2014/6

N2 - Let {Xn} be an integer-valued Markov chain with finite state space. Let Sn = Σk=0n Xk and let Ln(x) be the number of times Sk hits x ∈ ℤ up to step n. Define the normalized local time process ln(t,x) by (Formula presented.). The subject of this paper is to prove a functional weak invariance principle for the normalized sequence ln(t,x), i.e., we prove under the assumption of strong aperiodicity of the Markov chain that the normalized local times converge in distribution to the local time of the Brownian motion.

AB - Let {Xn} be an integer-valued Markov chain with finite state space. Let Sn = Σk=0n Xk and let Ln(x) be the number of times Sk hits x ∈ ℤ up to step n. Define the normalized local time process ln(t,x) by (Formula presented.). The subject of this paper is to prove a functional weak invariance principle for the normalized sequence ln(t,x), i.e., we prove under the assumption of strong aperiodicity of the Markov chain that the normalized local times converge in distribution to the local time of the Brownian motion.

KW - Brownian motion

KW - Local times

KW - Markov chains

KW - Weak invariance principle

UR - http://www.scopus.com/inward/record.url?scp=84899563282&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s10959-012-0438-z

DO - https://doi.org/10.1007/s10959-012-0438-z

M3 - مقالة

SN - 0894-9840

VL - 27

SP - 493

EP - 517

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 2

ER -

Weak Invariance Principle for the Local Times of Partial Sums of Markov Chains

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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