On convergence and convolutions of random signed measures
Let \(\mu_n\) be a sequence of random signed measures on the group \(G\) equal to either \(\mathbb{R}^d\) or the \(d\)-dimensional Torus \(T^d\). There are proved two general theorems stating that if \(\mu_n\) converges in some weak sense and a function \(K\) on \(G\) belongs to an appropriately chosen Sobolev or Besov space then the convolution \(\mu_n*K\) converges in law, in probability or almost surely in the Banach spaces \(C_0(G)\) or \(L^p(G)\). Using them convergence of distributional derivatives of \(\mu_n*K\) is also established. Examples for \(\mu_n\) covered are empirical processes (possibly arising from dependent data) and random signed measures defined by nonparametric kernel or wavelet based density estimators. An application of the general theorems to convolutions of density estimators is studied in detail. In particular, recent results of \textit{A. Schick} and \textit{W. Wefelmeyer} [J. Nonparametric Stat. 16, No. 6, 925--935 (2004; Zbl 1062.62065)] and \textit{E. Giné} and \textit{D. M. Mason} [Ann. Stat. 35, No. 3, 1105--1145 (2007; Zbl 1175.60017)] are extended and put into a general scheme.
scientific article; zbMATH DE number 5547309
scientific article