The Annals of Probability

A limit theorem for moments in space of the increments of Brownian local time

Simon Campese

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We prove a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. [Ann. Prob. 38 (2010) 396–438] and Rosen [Stoch. Dyn. 11 (2011) 5–48], which were later reproven by Hu and Nualart [Electron. Commun. Probab. 15 (2010) 396–410] and Rosen [In Séminaire de Probabilités XLIII (2011) 95–104 Springer] are included. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exclusively working in the space variable of the Brownian local time, which allows to give a unified argument for arbitrary orders. The main ingredients are Perkins’ semimartingale decomposition, the Kailath–Segall identity and an asymptotic Ray–Knight theorem by Pitman and Yor.

Article information

Ann. Probab., Volume 45, Number 3 (2017), 1512-1542.

Received: June 2015
Revised: January 2016
First available in Project Euclid: 15 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G44: Martingales with continuous parameter 60H05: Stochastic integrals

Kailath–Segall identity Brownian local time central limit theorem asymptotic Ray–Knight theorem


Campese, Simon. A limit theorem for moments in space of the increments of Brownian local time. Ann. Probab. 45 (2017), no. 3, 1512--1542. doi:10.1214/16-AOP1093.

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