Limit theorems and variation properties for fractional derivatives of the local time of a stable process
type: Item
instance of: scholarly article
title: Limit theorems and variation properties for fractional derivatives of the local time of a stable process
zbMATH Open document ID: 0749.60072
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1992-09-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77434
review text: Limit theorems for the occupation times of one-dimensional stable Markov processes \(X_ t\) of index \(\alpha\in]1,2]\) are considered. These results are refinements of the classical theorem of Darling and Kac, and they generalize results obtained by Yamada for Brownian motion. The resulting limit processes are fractional derivatives and Hilbert transforms of the stable local time. Namely, let \(f\in L^ 1_{loc}(\mathbb{R})\) be the one-side fractional derivative of \(g\), of order \(\gamma\), where \(0<\gamma<(\alpha-1)/2\) and \(g\) is a smooth function of compact support. Then as \(\lambda\to\infty\) \[ \lambda^{-(1- (1+\gamma)/\alpha)}\int_ 0^{\lambda t} f(X_ s)ds@>d>>\int g(x)dx\cdot H_ t^ 0, \] where \((H_ t^ 0)_{t\geq 0}\) is the fluctuating continuous additive functional of \(X\) defined by \[ H_ t^ 0={1 \over \Gamma(-\gamma)}\int_ 0^ \infty y^{-1-\gamma}[L_ t^{-y}-L_ t^ 0]dy, \] where \((L_ t^ x)_{t\geq 0}\) is local time at \(x\) for \(X\). The \(p\)-variation properties of these limit processes are also considered.
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 57286
zbMATH Keywords: occupation times
zbMATH Keywords: stable Markov processes
zbMATH Keywords: fractional derivatives
zbMATH Keywords: Hilbert transforms
author: Patrick J. Fitzsimmons
author: R. K. Getoor
reviewed by: Ilya S. Molchanov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 57286