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PreprintCC BY 4.0arXiv

Visiting time statistics

  • Maxim Kirsebom
  • Philipp Kunde
  • Tomas Persson
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Abstract

Many mixing dynamical systems (X,T,μ) are known to satisfy the hitting time statistics result limr → 0 μ{ x : τB(y,r) (x) > t/μ(B(y,r)) } = e−t, for μ-almost every y, where τB(y,r) (x) is the first hitting time of x to the ball B(y,r). Taking a different point of view, we fix x and consider τB(y,r) (x) as a function of y. We call this the visiting time of y from x, i.e. the time it takes for y to get a visit from x within a neighbourhood of radius r. We prove that limr → 0 μ{ y : τB(y,r) (x) > t/μ(B(y,r)) } = e−t, for μ-almost every x. As a byproduct we obtain a new method of proof for hitting time statistics.

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Cite

BibTeXRIS
BibTeX
@misc{kirsebom2026visiting,
  title = {{Visiting time statistics}},
  author = {Maxim Kirsebom and Philipp Kunde and Tomas Persson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.01453},
  eprint = {2608.01453},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.01453},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Visiting time statistics
AU  - Maxim Kirsebom
AU  - Philipp Kunde
AU  - Tomas Persson
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2608.01453
UR  - https://arxiv.org/abs/2608.01453
AB  - Many mixing dynamical systems (X,T,μ) are known to satisfy the hitting time statistics result lim_r → 0 μ{ x : τ_B(y,r) (x) > t/μ(B(y,r)) } = e^−t, for μ-almost every y, where τ_B(y,r) (x) is the first hitting time of x to the ball B(y,r). Taking a different point of view, we fix x and consider τ_B(y,r) (x) as a function of y. We call this the visiting time of y from x, i.e. the time it takes for y to get a visit from x within a neighbourhood of radius r. We prove that lim_r → 0 μ{ y : τ_B(y,r) (x) > t/μ(B(y,r)) } = e^−t, for μ-almost every x. As a byproduct we obtain a new method of proof for hitting time statistics.
ER  -