Limit theorems for local times of fractional Brownian motions and some other self-similar processes
type: Item
instance of: scholarly article
title: Limit theorems for local times of fractional Brownian motions and some other self-similar processes
zbMATH Open document ID: 0882.60037
author: Narn-Rueih Shieh
published in: Journal of Mathematics of Kyoto University
publication date: 1998-02-03T00:00:00Z
review text: Let \(X(t)\), \(t\geq 0\), be a real-valued \(H\)-self-similar process with stationary increments. When \(X(\cdot,\omega)\) has a regular local time \(L(t,x, \omega)\), \(t \geq 0\) and \(x\in\mathbb{R}\), the author considers the following four functionals \[ F_0(t,\omega)=L(t,0,\omega),\quad F_1(t,\omega)= \sup_x L(t,x,\omega), \] \[ F(\mu,t,\omega)= \int_{-\infty}^{\infty} L(t,x,\omega) \mu(dx),\quad F(f,t,\omega)=\int_{-\infty}^{\infty} L(t,x,\omega) f(x)dx, \] where \(\mu\) is a Borel measure and \(f\) is a Lebesgue integrable function on the line. He proves two limit theorems on the weak convergence in the space of continuous functions for \(F(\mu,t,\omega)\) and \(F(f,t,\omega)\), and if \(X\) is ergodic, he proves almost sure limit results for the logarithmic averages of \(F_i(t,\omega)\) (\(i=0,1\)). These results generalize those of \textit{D. A. Darling} and \textit{M. Kac} [Trans. Am. Math. Soc. 84, 444-458 (1957; Zbl 0078.32005)], \textit{T. Yamada} [J. Math. Kyoto Univ. 26, 309-332 (1986; Zbl 0618.60080)] and \textit{G. A. Brosamler} [Invent. Math. 20, 87-96 (1973; Zbl 0291.60037)] for one-dimensional Brownian motion. Some extensions of the results to the multiparameter and multidimensional cases are also given.
Mathematics Subject Classification ID: 60G18
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 1062458
zbMATH Keywords: self-similar process
zbMATH Keywords: local time
zbMATH Keywords: limit theorem
zbMATH Keywords: stationary increments
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518444
OpenAlex ID: W1556498707
DOI: 10.1215/KJM/1250518444
description: scientific article; zbMATH DE number 1062458