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

This is the authors' abstract. We don't add key points for this paper.

We proof a limit theorem for moments in space of the increments of Brownian time. As special cases for the second and third moments, previous results Chen et al. (Ann. Prob. 38, 2010, no. 1) and Rosen (Stoch. Dyn. 11, 2011,. 1), which were later reproven by Hu and Nualart (Electron. Commun. Probab.14, 2009; Electron. Commun. Probab. 15, 2010) and Rosen (S\\'eminaire de\\'es XLIII, Springer, 2011) are included. Furthermore, a conjecture Rosen for the fourth moment is settled. In comparison to the previous of proof, we follow a fundamentally different approach by exclusively in the space variable of the Brownian local time, which allows to give unified argument for arbitrary orders. The main ingredients are Perkins' decomposition, the Kailath-Segall identity and an asymptotic-Knight Theorem by Pitman and Yor.