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XXIX: 26, 266-289, LNM 1613 (1995)

**EISENBAUM, Nathalie**

Une version sans conditionnement du théorème d'isomorphisme de Dynkin (Limit theorems)

After establishing an unconditional version of Dynkin's isomorphism theorem, the author applies this theorem to give a new proof of Ray-Knight theorems for Brownian local times, and also to give another proof to limit theorems due to Rosen 2533 concerning the increments of the local times of a symmetric $\beta$-stable process for $\beta>1$. Some results by Marcus-Rosen (*Proc. Conf. Probability in Banach Spaces~8*, Birkhäuser 1992) on Laplace transforms of the increments of local time are extended

Comment: A general reference on the subject is Marcus-Rosen,*Markov Processes, Gaussian Processes, and Local Times*, Cambridge University Press (2006)

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

Nature: Original

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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

Nature: Original

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XLIV: 10, 207-213, LNM 2046 (2012)

**EISENBAUM, Nathalie**

Another failure in the analogy between Gaussian and semicircle laws (Non commutative probability theory)

Keywords: Gaussian law, Semicircle law, Free Poisson distribution, Free probability, Free convolution, $R$-transform, Dynkin isomorphism

Nature: Original

Une version sans conditionnement du théorème d'isomorphisme de Dynkin (Limit theorems)

After establishing an unconditional version of Dynkin's isomorphism theorem, the author applies this theorem to give a new proof of Ray-Knight theorems for Brownian local times, and also to give another proof to limit theorems due to Rosen 2533 concerning the increments of the local times of a symmetric $\beta$-stable process for $\beta>1$. Some results by Marcus-Rosen (

Comment: A general reference on the subject is Marcus-Rosen,

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

Nature: Original

Retrieve article from Numdam

XXXI: 20, 216-224, LNM 1655 (1997)

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

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

Nature: Original

Retrieve article from Numdam

XLIV: 10, 207-213, LNM 2046 (2012)

Another failure in the analogy between Gaussian and semicircle laws (Non commutative probability theory)

Keywords: Gaussian law, Semicircle law, Free Poisson distribution, Free probability, Free convolution, $R$-transform, Dynkin isomorphism

Nature: Original