## The Annals of Probability

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

Simon Campese

#### 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.

#### Article information

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

Dates
Revised: January 2016
First available in Project Euclid: 15 May 2017

https://projecteuclid.org/euclid.aop/1494835224

Digital Object Identifier
doi:10.1214/16-AOP1093

Mathematical Reviews number (MathSciNet)
MR3650408

Zentralblatt MATH identifier
1374.60019

#### Citation

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. https://projecteuclid.org/euclid.aop/1494835224

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