Seite - 392 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 Proof. Let >0begiven. BythesummabilityofK, thereexistsapositiverealnumber r, suchthat:∫ Rq−B(0,r) K(‖x‖)dx< with B(0,r) the open ball of radius r centered at the origin. Since B(0,r)⊂ B(0,rν−1) for ν > 0, thesameholds forallof the familyKν. LetB(0,M)beanopenball containingγ([0,1]). Then: μν ( R q−B(0,M+r) ) = ∫ Rq−B(0,M+r) ∫ 1 0 Kν(‖x−γ(t)‖)‖γ′(t)‖dtdx = ∫ 1 0 ∫ Rq−B(0,M+r) Kν(‖x−γ(t)‖)‖γ′(t)‖dxdt ≤ ∫ 1 0 ‖γ′(t)‖dt= l(γ) (8) where l(γ)denotes the lengthofγ. Thisproves the tightnessof the familyKν. Let f :Rq→Rbeaboundedcontinuousmapping. Itbecomes: Iν(f)= ∫ Rq ∫ 1 0 Kν(‖x−γ(t)‖) f(x)‖γ′(t)‖dtdx= ∫ 1 0 ∫ Rq Kν(‖x−γ(t)‖) f(x)‖γ′(t)‖dxdt = ∫ 1 0 ∫ Rq K(‖x‖) f (xν+γ(t))‖γ′(t)‖dxdt (9) andsince f isbounded, thedominatedconvergence theoremshowsthat: lim ν→0 Iν(f)= ∫ 1 0 f (γ(t))‖γ′(t)‖dt provingthesecondpartof theclaim. Thedensity in (7) is forasinglecurveof the formd(x)= l(γ)−1 ∫1 0 K(‖x−γ(t)‖)‖γ′(t)‖dtwith l(γ) the lengthof thecurveγ. It is invariantunder thechangeof theparameterandcanbewritten ina moreconcisewayas: ∫ 1 0 K(‖x−γ(η)‖)‖dη (10) whereη is thearclength times l(γ)−1. This formallowsasimpleprobabilistic interpretationof thedensityd: if apointu isdrawnonthe curveγaccordingtoauniformdistributionandindependentlyavectorv inRqwithadensityK (the multivariatekernelcorrespondingtoK), thenthedensityofx=u+v isgivenbyEquation(10). Proposition 2. If themultivariate kernelK has a finite secondmoment, that is the univariate kernel K is such that: M= ∫ R+ rq+1K(r)dr<+∞ then theWassersteindistancebetween thedensities d1,d2 associatedwith smoothcurvesγ1,γ2 is boundedby: 2Vol(Sq−1)M+D(γ1,γ2) with : D(γ1,γ2)= ∫ 1 0 ‖γ1(η)−γ2(η)‖2dη where eachcurve isparametrizedby the scaledarclengthas in (10). 392