Limit theorems for general functionals of Brownian local times

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 [2, 3, 4, 5, 10, 13, 14], which delved into special cases of the considered problem. Notably, [3, 4, 5, 13, 14] explored quadratic and cubic cases, predominantly utilizing the method of moments technique, Malliavin calculus and Ray-Knight theorems to demonstrate asymptotic mixed normality. Meanwhile, [2] extended the theory to general polynomials under a non-standard centering by exploiting Perkins’ semimartingale representation of local time and the Kailath-Segall formula. In contrast to the methodologies employed in [3, 4, 5, 13], our approach relies on infill limit theory for semimartingales, as formulated in [6, 8]. Notably, 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.