Page - 240 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 wherevk∈Rqk isavectorcorrespondingto theWk-componentofξ. Ifξ∈P∗V,wesee from(19) that ψk(ξ) ispositivedefinite. In thiscase,wehave: φk(ξ)= ( 1 tvkψk(ξ)−1 Iqk )( ξkk− tvkψk(ξ)−1vk ψk(ξ) )( 1 ψk(ξ) −1vk Iqk ) , (21) so thatwegetageneralizationof (18), that is, (φk(ξ) −1)11=(ξkk− tvkψk(ξ)−1vk)−1. (22) Ontheotherhand, ifqk=0, thenφk(ξ)−1= ξ−1kk . Weremarkthatψ1(ξ)=ψ(ξ′), andthatsomepartof theargumentabove isparallel to theproof ofTheorem1. Theorem2. TheKoszul–Vinbergcharacteristic functionϕV ofP∗V isgivenby the following formula: ϕV(ξ)=CV r ∏ k=1 ( φk(ξ) −1)1+qk/2 11 ∏ qk>0 (detψk(ξ)) −1/2 (ξ∈P∗V), (23) whereCV :=(2π)(N−r)/2∏rk=1Γ(1+ qk 2 )andN :=dimZV. Proof. Weshall showthe statementby inductionon the rankas in theproof ofTheorem1. Then, it suffices toshowthat: ϕV(ξ)=(2π)q1/2Γ(1+ q1 2 )(φ1(ξ) −1)1+q1/211 (detψ1(ξ)) −sgn(q1)/2ϕV′(ξ′) (24) forξ∈P∗V as in (10),where (detψ1(ξ))−sgn(q1)/2 is interpretedas: (detψ1(ξ))−sgn(q1)/2 := { 1 (q1=0), (detψ1(ξ))−1/2 (q1>0). Whenq1=0,wehave: ϕV(ξ)= ∫ ∞ 0 ∫ PV′ e−x11ξ11e−(x ′|ξ′)dx11dx′ = ξ−111 ϕV′(ξ ′), whichmeans (24). When q1> 0, theEuclideanmeasure dx equals 2q1/2x q1 11dx11du˜dx˜ ′ by the changeof variables in (6). Indeed, thecoefficient2q1/2 comes fromthenormalizationof the innerproductonW Rq1 regardedasasubspaceofZV. Then,wehaveby(15): ϕV(ξ)= ∫ ∞ 0 ∫ R q1 ∫ PV′ e−x11(ξ11− tvψ(ξ′)−1v)e−x11 t(u˜+ψ(ξ′)−1v)ψ(ξ′)(u˜+ψ(ξ′)−1v)e−(x˜ ′|ξ′) ×2q1/2xq111dx11du˜dx˜′. By theGaussian integral formula,wehave:∫ R q1 e−x11 t(u˜+ψ(ξ′)−1v)ψ(ξ′)(u˜+ψ(ξ′)−1v)du˜=πq1/2x−q1/211 (detψ(ξ ′))−1/2. 240