Limit theorems for local times of fractional Brownian motions and some other self-similar processes
Let \(X(t)\), \(t\geq 0\), be a real-valued \(H\)-self-similar process with stationary increments. When \(X(\cdot,\omega)\) has a regular local time \(L(t,x, \omega)\), \(t \geq 0\) and \(x\in\mathbb{R}\), the author considers the following four functionals \[ F_0(t,\omega)=L(t,0,\omega),\quad F_1(t,\omega)= \sup_x L(t,x,\omega), \] \[ F(\mu,t,\omega)= \int_{-\infty}^{\infty} L(t,x,\omega) \mu(dx),\quad F(f,t,\omega)=\int_{-\infty}^{\infty} L(t,x,\omega) f(x)dx, \] where \(\mu\) is a Borel measure and \(f\) is a Lebesgue integrable function on the line. He proves two limit theorems on the weak convergence in the space of continuous functions for \(F(\mu,t,\omega)\) and \(F(f,t,\omega)\), and if \(X\) is ergodic, he proves almost sure limit results for the logarithmic averages of \(F_i(t,\omega)\) (\(i=0,1\)). These results generalize those of \textit{D. A. Darling} and \textit{M. Kac} [Trans. Am. Math. Soc. 84, 444-458 (1957; Zbl 0078.32005)], \textit{T. Yamada} [J. Math. Kyoto Univ. 26, 309-332 (1986; Zbl 0618.60080)] and \textit{G. A. Brosamler} [Invent. Math. 20, 87-96 (1973; Zbl 0291.60037)] for one-dimensional Brownian motion. Some extensions of the results to the multiparameter and multidimensional cases are also given.
scientific article; zbMATH DE number 1062458