On Convergence and Convolutions of Random Signed Measures

Richard Nickl ()
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Richard Nickl: University of Connecticut

Journal of Theoretical Probability, 2009, vol. 22, issue 1, 38-56

Abstract: Abstract Let μ n be a sequence of random finite signed measures on the locally compact group G equal to either $\mathbb{T}^{d}$ or ℝ d . We give weak conditions on the sequence μ n and on functions K such that the convolution product μ n *K, and its derivatives, converge in law, in probability, or almost surely in the Banach spaces $\mathsf{C}_{0}(G)$ or L p (G). Examples for sequences μ n covered are the empirical process (possibly arising from dependent data) and also random signed measures $\sqrt{n}(\tilde{\mathbb{P}}_{n}-\mathbb{P})$ where $\tilde{\mathbb{P}}_{n}$ is some (nonparametric) estimator for the measure ℙ, including the usual kernel and wavelet based density estimators with MISE-optimal bandwidths. As a statistical application, we apply the results to study convolutions of density estimators.

Keywords: Convolutions; Limit theorems; Banach space; Density estimator; 60F17; 28A33; 60B15; 46E27 (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10959-008-0177-3

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