On a completeness problem in a Fourier-based probability metrics in ℝ^N

2016/09/01 by Stawiska, Małgorzata
#33C15 #60B10 #60E10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1609.00343

Abstract

We study completeness of the spaces P_s^= of probability measures in ℝ^N which have equal (prescribed) moments up to order s ∈ ℕ, endowed with the metric d_s(μ,ν)=sup_{x ∈ ℝ^N∖ 0}(| μ(x)- ν(x)|)/(|x|^s), where μ is the characteristic function of μ. We prove that the spaces (P_s^=,d_s) are complete if s is even and construct suitable counterexamples to completeness for all odd s. This solves an open problem formulated by J. Carrillo and G. Toscani in 2007.

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