Random walk local time approximated by a Brownian sheet combined with an independent Brownian motion
Let \(\xi(k,n)\), \(n\geq 1\), \(k\in \mathbb{Z}\), be a local time of a symmetric simple random walk on a line, or, in other words, a number of excursions away from \(k\) completed before \(n\). The paper is devoted to the strong approximations of centered local time \(\xi(k,n)-\xi(0,n)\) in terms of a Brownian sheet and an independent Wiener process, time changed by an independent Brownian local time. Some consequences such as week convergence and the laws of iterated logarithm (related to \(\xi(k,n)\)) are established. The paper also contains a survey and an extensive list of references concerning the results on such strong approximations of the local time.
scientific article; zbMATH DE number 5598059
scientific article