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XXXI: 20, 216-224, LNM 1655 (1997)
EISENBAUM, Nathalie
Théorèmes limites pour les temps locaux d'un processus stable symétrique (Limit theorems)
Using Dynkin's isomorphism, a central-limit type theorem is derived for the local times of a stable symmetric process of index $\beta$ at a finite number $n$ of levels. The limiting process is expressed in terms of a fractional, $n$-dimensional Brownian sheet with Hurst index $\beta-1$. The case when $n=1$ is due to Rosen 2533, and, for Brownian local times, to Yor 1709
Comment: This kind of result is now understood as a weak form of theorems à la Ray-Knight, describing the local times of a stable symmetric process: see Eisenbaum-Kaspi-Marcus-Rosen-Shi Ann. Prob. 28 (2000) for a Ray-Knight theorem involving fractional Brownian motion. Marcus-Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press (2006) is a general reference on the subject
Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet
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