Małgorzata Stawiska ^[a]

On a completeness problem in Fourier-based probability metrics in \(\mathbb{R}^N\)

Pages: 263-282
Received: 23 April 2019
Accepted in revised form: 18 September 2019
Mathematics Subject Classification (2010): Primary 60B10; Secondary 60E10, 33C15.
Keywords: Convergence of probability measures; characteristic function; Fourier transform; moments; completeness; exponential integral function.
Author address:
[a]: Mathematical Reviews, 416 Fourth St., Ann Arbor, MI 48103, USA

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Abstract: We study completeness of the spaces \(\mathcal{P}_s^=\) of probability measures in \(\mathbb{R}^N\) which have equal (prescribed) moments up to order \(s \in \mathbb{N}\), endowed with the metric \(d_s(\mu,\nu)=\sup_{x \in \mathbb{R}^N\setminus 0}\frac{|\hat \mu(x)-\hat \nu(x)|}{|x|^s}\), where \(\hat \mu\) is the characteristic function of \(\mu\). We prove that the spaces \((\mathcal{P}_s^=,d_s)\) are complete if \(s\) is even and construct counterexamples to completeness for all odd \(s\). This solves an open problem formulated by J. Carrillo and G. Toscani in 2007 ([CT]).