The favorite point of a Poisson process
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.
scientific article; zbMATH DE number 757539