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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: 12, 309-325, LNM 2006 (2011)

**TUDOR, Ciprian A.**

Asymptotic Cramér's theorem and analysis on Wiener space (Limit theorems, Stochastic analysis)

Keywords: Multiple stochastic integrals, Limit theorems, Malliavin calculus, Stein's method

Nature: Original

XLIV: 08, 167-190, LNM 2046 (2012)

**HAJRI, Hatem**

Discrete approximation to solution flows of Tanaka's SDE related to Walsh Brownian motion (Stochastic calculus, Limit theorems)

Keywords: Walsh's Brownian motion, Tanaka's SDE, Local times

Nature: Original

XLIV: 12, 247-269, LNM 2046 (2012)

**MARTY, Renaud**; **SØLNA, Knut**

Asymptotic behavior of oscillatory fractional processes (Theory of processes, Limit theorems)

Keywords: Fractional processes, Brownian motion, Waves in random media

Nature: Original

XLIV: 17, 375-399, LNM 2046 (2012)

**HARRIS, Simon C.**; **ROBERTS, Matthew I.**

Branching Brownian motion: Almost sure growth along scaled paths (Limit theorems, Theory of processes)

Keywords: Branching Brownian motion

Nature: Original

XLIV: 19, 409-428, LNM 2046 (2012)

**BERCU, Bernard**; **BONY, Jean-François**; **BRUNEAU, Vincent**

Large deviations for Gaussian stationary processes and semi-classical analysis (Limit theorems, theory of processes)

Keywords: Large deviations, Gaussian processes, Toeplitz matrices, Distribution of eigenvalues

Nature: Original

XLV: 01, 3-89, LNM 2078 (2013)

**NOURDIN, Ivan**

Lectures on Gaussian Approximations with Malliavin Calculus (Limit Theorems)

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: 12, 309-325, LNM 2006 (2011)

Asymptotic Cramér's theorem and analysis on Wiener space (Limit theorems, Stochastic analysis)

Keywords: Multiple stochastic integrals, Limit theorems, Malliavin calculus, Stein's method

Nature: Original

XLIV: 08, 167-190, LNM 2046 (2012)

Discrete approximation to solution flows of Tanaka's SDE related to Walsh Brownian motion (Stochastic calculus, Limit theorems)

Keywords: Walsh's Brownian motion, Tanaka's SDE, Local times

Nature: Original

XLIV: 12, 247-269, LNM 2046 (2012)

Asymptotic behavior of oscillatory fractional processes (Theory of processes, Limit theorems)

Keywords: Fractional processes, Brownian motion, Waves in random media

Nature: Original

XLIV: 17, 375-399, LNM 2046 (2012)

Branching Brownian motion: Almost sure growth along scaled paths (Limit theorems, Theory of processes)

Keywords: Branching Brownian motion

Nature: Original

XLIV: 19, 409-428, LNM 2046 (2012)

Large deviations for Gaussian stationary processes and semi-classical analysis (Limit theorems, theory of processes)

Keywords: Large deviations, Gaussian processes, Toeplitz matrices, Distribution of eigenvalues

Nature: Original

XLV: 01, 3-89, LNM 2078 (2013)

Lectures on Gaussian Approximations with Malliavin Calculus (Limit Theorems)

Nature: Original