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May 2017 A limit theorem for moments in space of the increments of Brownian local time
Simon Campese
Ann. Probab. 45(3): 1512-1542 (May 2017). DOI: 10.1214/16-AOP1093

Abstract

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.

Citation

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Simon Campese. "A limit theorem for moments in space of the increments of Brownian local time." Ann. Probab. 45 (3) 1512 - 1542, May 2017. https://doi.org/10.1214/16-AOP1093

Information

Received: 1 June 2015; Revised: 1 January 2016; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 1374.60019
MathSciNet: MR3650408
Digital Object Identifier: 10.1214/16-AOP1093

Subjects:
Primary: 60F05 , 60G44 , 60H05

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

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 3 • May 2017
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