The favorite point of a Poisson process
type: Item
instance of: scholarly article
title: The favorite point of a Poisson process
zbMATH Open document ID: 0824.60042
DOI: 10.1016/0304-4149(94)00077-7
author: D. Khoshnevisan
author: Thomas M. Lewis
published in: Stochastic Processes and their Applications
publication date: 1995-05-23T00:00:00Z
review text: Let \(N\) be a Poisson process with the unit rate, \(\xi_ t (x)\) be the occupation of \(x\) by time \(t\), i.e. \(\xi_ t (x) = \int^ t_ 0 I_{\{x\}} (N_ u) du\). The favorite point of \(N\) up to time \(t\) is defined as follows: \[ X_ t = \min \bigl \{k \geq 0 : \xi_ t (k) \geq \xi_ t (i) \text{ for all } i \geq 0 \bigr\}. \] The paper contains a number of results concerning the asymptotic behavior of \(X_ t\) when \(t \to \infty\). In particular, it is proved that the law of iterated logarithm \(\lim \sup (X_ t - t)/ \sqrt {2t \ln nt} = 1\) a.s. holds in spite of the fact that \((X_ t - t)/ \sqrt t\) does not converge to the normal law.
reviewed by: Yuri Kabanov
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 757539
zbMATH Keywords: Poisson process
zbMATH Keywords: occupation time
zbMATH Keywords: most visited site
zbMATH Keywords: law of iterated logarithm
zbMATH Keywords: Dirichlet process
MaRDI profile type: Publication
full work available at URL: 0304-4149(94)00077-7
OpenAlex ID: W2053199520
cites work: Q5520081
cites work: A theorem of Feller revisited
cites work: The most visited site of Brownian motion and simple random walk
cites work: On the law of the iterated logarithm
cites work: Q3723443
cites work: Association of Random Variables, with Applications
cites work: Q5624436
cites work: Q3923307
cites work: A note on the Borel-Cantelli lemma
cites work: General one-sided laws of the iterated logarithm
cites work: Q5521185
cites work: Q5643431
cites work: The asymptotic behavior of spacings under Kakutani's model for interval subdivision
cites work: Entropy and maximal spacings for random partitions
description: scientific article; zbMATH DE number 757539