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Limit theorems and variation properties for fractional derivatives of the local time of a stable process. (English) Zbl 0749.60072
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.

MSC:
60J55 Local time and additive functionals
60J99 Markov processes
60F05 Central limit and other weak theorems
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