In this paper we study the asymptotic behavior of additive functionals of Brownian motion which are not necessarily of bounded variation. The result is then applied to the Hilbert transform of the Brownian local time.
Masayoshi Takeda. Tusheng Zhang. "Asymptotic properties of additive functionals of Brownian motion." Ann. Probab. 25 (2) 940 - 952, April 1997. https://doi.org/10.1214/aop/1024404425