Home > Research > Publications & Outputs > Strong law of large numbers for a function of t...

Research

Associated organisational unit

Electronic data

  • Intersections_JoTP

    Accepted author manuscript, 167 KB, PDF document

    Available under license: CC BY: Creative Commons Attribution 4.0 International License

Links

Text available via DOI:

Keywords

View graph of relations

Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published

Standard

Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$. / Asymont, Inna; Korshunov, Dmitry.
In: Journal of Theoretical Probability, Vol. 33, 01.12.2020, p. 2315–2336.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Asymont, I & Korshunov, D 2020, 'Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$', Journal of Theoretical Probability, vol. 33, pp. 2315–2336. https://doi.org/10.1007/s10959-019-00937-6

APA

Asymont, I., & Korshunov, D. (2020). Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$. Journal of Theoretical Probability, 33, 2315–2336. https://doi.org/10.1007/s10959-019-00937-6

Vancouver

Asymont I, Korshunov D. Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$. Journal of Theoretical Probability. 2020 Dec 1;33:2315–2336. Epub 2019 Aug 28. doi: 10.1007/s10959-019-00937-6

Author

Asymont, Inna ; Korshunov, Dmitry. / Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$. In: Journal of Theoretical Probability. 2020 ; Vol. 33. pp. 2315–2336.

Bibtex

@article{66ad6ca904e2462da2f72ff65cc5d773,
title = "Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$",
abstract = "For an arbitrary transient random walk (Sn)n≥0 in Zd, d≥1, we prove a strong law of large numbers for the spatial sum ∑x∈Zd f(l(n,x)) of a function f of the local times l(n,x)=∑ni=0 II{Si=x}. Particular cases are the number of (a)visited sites [first considered by Dvoretzky and Erd{\H o}s (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function f(i)= II{i≥1};(b)α-fold self-intersections of the random walk [studied by Becker and K{\"o}nig (J Theor Probab 22:365–374, 2009)], which corresponds to f(i)=iα;(c)sites visited by the random walk exactly j times [considered by Erd{\H o}s and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where f(i)=II{i=j}.",
keywords = "Transient random walk in Zd, Local times, Strong law of large numbers",
author = "Inna Asymont and Dmitry Korshunov",
year = "2020",
month = dec,
day = "1",
doi = "10.1007/s10959-019-00937-6",
language = "English",
volume = "33",
pages = "2315–2336",
journal = "Journal of Theoretical Probability",
issn = "0894-9840",
publisher = "Springer New York",

}

RIS

TY - JOUR

T1 - Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$

AU - Asymont, Inna

AU - Korshunov, Dmitry

PY - 2020/12/1

Y1 - 2020/12/1

N2 - For an arbitrary transient random walk (Sn)n≥0 in Zd, d≥1, we prove a strong law of large numbers for the spatial sum ∑x∈Zd f(l(n,x)) of a function f of the local times l(n,x)=∑ni=0 II{Si=x}. Particular cases are the number of (a)visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function f(i)= II{i≥1};(b)α-fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to f(i)=iα;(c)sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where f(i)=II{i=j}.

AB - For an arbitrary transient random walk (Sn)n≥0 in Zd, d≥1, we prove a strong law of large numbers for the spatial sum ∑x∈Zd f(l(n,x)) of a function f of the local times l(n,x)=∑ni=0 II{Si=x}. Particular cases are the number of (a)visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function f(i)= II{i≥1};(b)α-fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to f(i)=iα;(c)sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where f(i)=II{i=j}.

KW - Transient random walk in Zd

KW - Local times

KW - Strong law of large numbers

U2 - 10.1007/s10959-019-00937-6

DO - 10.1007/s10959-019-00937-6

M3 - Journal article

VL - 33

SP - 2315

EP - 2336

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

ER -