The escape rate of favorite sites of simple random walk and brownian motion

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The escape rate of favorite sites of simple random walk and brownian motion. / Lifshits, Mlkhail A.; Shi, Zhan.

In: Annals of Probability, Vol. 32, No. 1 A, 01.01.2004, p. 129-152.

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

BibTeX

@article{a60a18d7c0b04986b5c5fad3e32d0c8b,

title = "The escape rate of favorite sites of simple random walk and brownian motion",

abstract = "Consider a simple symmetric random walk on the integer lattice Z. For each n, let V(n) denote a favorite site (or most visited site) of the random walk in the first n steps. A somewhat surprising theorem of Bass and Griffin [Z. Wahrsch. Verw. Gebiete 70 (1985) 417-436] says that V is almost surely transient, thus disproving a previous conjecture of Erd{\"o}s and R{\'e}v{\'e}sz [Mathematical Structures-Computational MathematicsMathematical Modeling 2 (1984) 152-157]. More precisely, Bass and Griffin proved that almost surely, lim inf n→ |V(n)/n| 1/2(logn) -γ 0 if γ < 1, and is infinity if γ > 11 (eleven). The present paper studies the rate of escape of V(n). We show that almost surely, the {"}lim inf{"} expression in question is 0 if γ 1, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.",

keywords = "Brownian motion, Favorite site, Local time, Random walk",

author = "Lifshits, {Mlkhail A.} and Zhan Shi",

year = "2004",

month = jan,

day = "1",

language = "English",

volume = "32",

pages = "129--152",

journal = "Annals of Probability",

issn = "0091-1798",

publisher = "Institute of Mathematical Statistics",

number = "1 A",

}

RIS

TY - JOUR

T1 - The escape rate of favorite sites of simple random walk and brownian motion

AU - Lifshits, Mlkhail A.

AU - Shi, Zhan

PY - 2004/1/1

Y1 - 2004/1/1

N2 - Consider a simple symmetric random walk on the integer lattice Z. For each n, let V(n) denote a favorite site (or most visited site) of the random walk in the first n steps. A somewhat surprising theorem of Bass and Griffin [Z. Wahrsch. Verw. Gebiete 70 (1985) 417-436] says that V is almost surely transient, thus disproving a previous conjecture of Erdös and Révész [Mathematical Structures-Computational MathematicsMathematical Modeling 2 (1984) 152-157]. More precisely, Bass and Griffin proved that almost surely, lim inf n→ |V(n)/n| 1/2(logn) -γ 0 if γ < 1, and is infinity if γ > 11 (eleven). The present paper studies the rate of escape of V(n). We show that almost surely, the "lim inf" expression in question is 0 if γ 1, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.

AB - Consider a simple symmetric random walk on the integer lattice Z. For each n, let V(n) denote a favorite site (or most visited site) of the random walk in the first n steps. A somewhat surprising theorem of Bass and Griffin [Z. Wahrsch. Verw. Gebiete 70 (1985) 417-436] says that V is almost surely transient, thus disproving a previous conjecture of Erdös and Révész [Mathematical Structures-Computational MathematicsMathematical Modeling 2 (1984) 152-157]. More precisely, Bass and Griffin proved that almost surely, lim inf n→ |V(n)/n| 1/2(logn) -γ 0 if γ < 1, and is infinity if γ > 11 (eleven). The present paper studies the rate of escape of V(n). We show that almost surely, the "lim inf" expression in question is 0 if γ 1, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.

KW - Brownian motion

KW - Favorite site

KW - Local time

KW - Random walk

UR - http://www.scopus.com/inward/record.url?scp=2142774188&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:2142774188

VL - 32

SP - 129

EP - 152

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 1 A

ER -

ID: 37010783