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XXV: 33, 407-424, LNM 1485 (1991)

**ROSEN, Jay S.**

Second order limit laws for the local times of stable processes (Limit theorems)

Using the method of moments, a central limit theorem is established for the increments $L^x_t-L^0_t$ of the local times of a symmetric $\beta$-stable process ($\beta>1$). The limit law is that of a fractional Brownian sheet, with Hurst index $\beta-1$, time-changed via $L_t^0$ in its time variable

Comment: Another proof due to Eisenbaum 3120 uses Dynkin's isomorphism. Ray-Knight theorems for these local times can be found in Eisenbaum-Kaspi-Marcus-Rosen-Shi*Ann. Prob.* **28** (2000). A good reference on this subject is Marcus-Rosen, *Markov Processes, Gaussian Processes, and Local Times*, Cambridge University Press (2006)

Keywords: Local times, Stable processes, Method of moments, Fractional Brownian motion, Brownian sheet

Nature: Original

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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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XLIII: 01, 3-70, LNM 2006 (2011)

**PICARD, Jean**

Representation formulae for the fractional Brownian motion (Theory of processes)

Keywords: Fractional Brownian motion, Brownian motion

Nature: Original, Survey

XLIII: 08, 215-219, LNM 2006 (2011)

**PRATELLI, Maurizio**

A Remark on the $1/H$-variation of the Fractional Brownian Motion (Theory of processes)

Keywords: Fractional Brownian motion, $p$-variation, Ergodic theorem

Nature: Exposition

XLIII: 09, 221-239, LNM 2006 (2011)

**MAROUBY, Matthieu**

Simulation of a Local Time Fractional Stable Motion (Theory of processes)

Keywords: Stable processes, Self-similar processes, Shot noise series, Local times, Fractional Brownian motion, Simulation

Nature: Original

XLV: 15, 365-400, LNM 2078 (2013)

**PAGÈS, Gilles**

Functional Co-monotony of Processes with Applications to Peacocks and Barrier Options (Theory of processes)

Keywords: Co-monotony, antithetic simulation method, processes with independent increments, Liouville processes, fractional Brownian motion, Asian options, sensitivity, barrier options

Nature: Original

Second order limit laws for the local times of stable processes (Limit theorems)

Using the method of moments, a central limit theorem is established for the increments $L^x_t-L^0_t$ of the local times of a symmetric $\beta$-stable process ($\beta>1$). The limit law is that of a fractional Brownian sheet, with Hurst index $\beta-1$, time-changed via $L_t^0$ in its time variable

Comment: Another proof due to Eisenbaum 3120 uses Dynkin's isomorphism. Ray-Knight theorems for these local times can be found in Eisenbaum-Kaspi-Marcus-Rosen-Shi

Keywords: Local times, Stable processes, Method of moments, Fractional Brownian motion, Brownian sheet

Nature: Original

Retrieve article from Numdam

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

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

XLIII: 01, 3-70, LNM 2006 (2011)

Representation formulae for the fractional Brownian motion (Theory of processes)

Keywords: Fractional Brownian motion, Brownian motion

Nature: Original, Survey

XLIII: 08, 215-219, LNM 2006 (2011)

A Remark on the $1/H$-variation of the Fractional Brownian Motion (Theory of processes)

Keywords: Fractional Brownian motion, $p$-variation, Ergodic theorem

Nature: Exposition

XLIII: 09, 221-239, LNM 2006 (2011)

Simulation of a Local Time Fractional Stable Motion (Theory of processes)

Keywords: Stable processes, Self-similar processes, Shot noise series, Local times, Fractional Brownian motion, Simulation

Nature: Original

XLV: 15, 365-400, LNM 2078 (2013)

Functional Co-monotony of Processes with Applications to Peacocks and Barrier Options (Theory of processes)

Keywords: Co-monotony, antithetic simulation method, processes with independent increments, Liouville processes, fractional Brownian motion, Asian options, sensitivity, barrier options

Nature: Original