  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.
