Page - 239 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 Assume q1> 0. Keeping inmindthatW˜1 R⊕W andW Rq1 by (4),wehave for ξ∈ZV as in (10), φ1(ξ)= ( ξ11 tv v ψ(ξ′) ) ∈Sym(1+q1,R), (12) wherev=vect(V)∈Rqi andψ :ZV′→Sym(qi,R) isdefinedinsuchawaythat: (UtU|ξ′)= tuψ(ξ′)u (ξ′ ∈ZV′,U∈W, u=vect(U)∈Rq1). (13) Ontheotherhand, forx∈ZV as in (6),wehave: (x|ξ)= x11ξ11+2x11tu˜v+x11tu˜ψ(ξ′)u˜+(x˜′|ξ). (14) OwingtoLemma1(iii), theelementξ∈ZV belongs toP∗V if andonly if theright-handside is strictly positive forallx11≥0, U˜∈W and x˜′ ∈PV′ with (x11, x˜′) =(0,0). Assumeξ∈P∗V. Consideringthe casex11=0,wehave (x˜′|ξ′)>0 forall x˜′ ∈PV′\{0},whichmeans thatξ′ ∈P∗V′. Then, thequantity in (13) is strictlypositive fornon-zeroU becauseUtU belongs toPV \{0}. Thus, ψ(ξ′) is positive definite,and(14) is rewrittenas: (x|ξ)= x11(ξ11− tvψ(ξ′)−1v)+x11t(u˜+ψ(ξ′)−1v)ψ(ξ′)(u˜+ψ(ξ′)−1v)+(x˜′|ξ′). (15) Therefore,weobtain: P∗V= { ξ∈ZV ; ξ′ ∈P∗V′ andξ11− tvψ(ξ′)−1v>0 } . (16) Ontheotherhand,wesee from(12) that: φ1(ξ)= ( 1 tvψ(ξ′)−1 Iq1 )( ξ11− tvψ(ξ′)−1v ψ(ξ′) )( 1 ψ(ξ′)−1v Iq1 ) . (17) Hence,wededuce (11) from(16)and(17). Wenote that, ifq1>0, the (1,1)-componentof the inversematrixφ1(ξ)−1 isgivenby: (φ1(ξ) −1)11=(ξ11− tvψ(ξ′)−1v)−1 (18) thanks to (17). 3.Koszul–VinbergCharacteristicFunctionofP∗V Wedenote by ϕV theKoszul–Vinberg characteristic function ofP∗V. In this section, we give anexplicit formulaofϕV. Recall thatthelinearmapψ :ZV′→Sym(q1,R)playsanimportantrole intheproofofTheorem1. Weshall introducesimilar linearmapsψk :ZV→ Sym(qk,R) for k such that qk> 0. LetWkbe the subspaceof W˜k consistingofW ∈ W˜k forwhichw1 = xkk = 0. Then, clearly,Wk ∑⊕l>kVlk and dimWk= qk. Ifqk>0,usingthesameorthogonalbasisofVlk as in theprevioussection,wehave the isomorphismWk W →w=vect(W)∈Rqk. Similarly to (8),wedefineψkby: (WtW|ξ)= twψk(ξ)w (ξ∈ZV, W∈Wk,w=vect(W)∈Rqk). (19) Then,wehave: φk(ξ)= ( ξkk tvk vk ψk(ξ) ) (ξ∈ZV), (20) 239