  In this paper, we present the asymptotic theory for integrated functions of
increments of Brownian local times in space. Specifically, we determine their
first-order limit, along with the asymptotic distribution of the fluctuations.
Our key result establishes that a standardized version of our statistic
converges stably in law towards a mixed normal distribution. Our contribution
builds upon a series of prior works by S. Campese, X. Chen, Y. Hu, W.V. Li,
M.B. Markus, D. Nualart and J. Rosen \cite{C17,CLMR10,HN09,HN10,MR08,R11,R11b},
which delved into special cases of the considered problem, such as quadratic,
cubic and polynomial cases. We establish the limit theorem for general
functions that satisfy mild smoothness and growth conditions. This extends the
scope beyond the polynomial cases studied in previous works, providing a more
comprehensive understanding of the asymptotic properties of the considered
functionals.
