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

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.