Some asymptotic properties of the local time of the uniform empirical process
Let \(U_1,U_2, \dots\) denote a sequence of independent random variables, uniformly distributed on \([0,1]\), and let \(\alpha_n(t): =\sqrt n(F_n (t)-t)\), \(0\leq t\leq 1\), denote the uniform empirical process. The authors study level crossings of \(\alpha_n\), defined by \[ L_1^x(\alpha_n) =n^{-1/2} \sum_{0\leq t\leq 1}\mathbf{1}_{\{\alpha_n(t) =x\}},\quad x\in\mathbb{R}. \] The main results of the present paper are LIL-type theorems, concerning the almost sure asymptotic behavior of various functionals of \(L_1^x(\alpha_n)\), \(x\in\mathbb{R}\). These functionals include \((L_1^x (\alpha_n)\), \(\sup_x L_1^x(\alpha_n))\) and \(\int(L^x (\alpha_n))^pdx\), \(p\in\mathbb{N}\). The proofs are based on a coupling of the uniform empirical process and Brownian bridge together with tail estimates for functionals of the local time of Brownian bridge.
scientific article; zbMATH DE number 1395962