## Abstract

For an arbitrary transient random walk (Sn)n≥0 in Z^{d}, d≥ 1 , we prove a strong law of large numbers for the spatial sum ∑x∈Zdf(l(n,x)) of a function f of the local times l(n,x)=∑i=0nI{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) = I{ 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) = i^{α};(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) = I{ i= j}.

Original language | English |
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Pages (from-to) | 2315-2336 |

Number of pages | 22 |

Journal | Journal of Theoretical Probability |

Volume | 33 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Dec 2020 |

Externally published | Yes |

## Keywords

- Local times
- Strong law of large numbers
- Transient random walk in Z
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