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

TL;DR

This paper establishes a new limit theorem for the moments in space of Brownian local time increments, unifying previous results for specific moments and confirming Rosen's conjecture for the fourth moment using a novel space-variable approach.

Contribution

It introduces a fundamentally different method working in the space variable, providing a unified proof for moments of all orders and settling a conjecture for the fourth moment.

Findings

Proved a limit theorem for moments in space of Brownian local time increments.

Unified previous results for second and third moments.

Confirmed Rosen's conjecture for the fourth moment.

Abstract

We proof 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, no. 1) and Rosen (Stoch. Dyn. 11, 2011, no. 1), which were later reproven by Hu and Nualart (Electron. Commun. Probab. 14, 2009; Electron. Commun. Probab. 15, 2010) and Rosen (S\'eminaire de Probabilit\'es XLIII, Springer, 2011) 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.