Page - 354 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 wheredY isaLebesguemeasureonthecoefficientsofY. Let p˜=ψ(K×a+). λ∈Λ+nevervanishes ona+ andp\ p˜hasanullmeasure. Thus, ∫ p˜ t(Y)∏ λ∈Λ+ 1 λ(HY) dY= c1 ∫ K , ∫ a+ t(Adk(H))dkdH, (12) whereHY is thepoint ina+ suchthat thereexistsk inK suchthatψ(k,HY)=Y. Calculation inp. 73 in [27]alsogives that forall t∈Cc(p), thereexists c2>0, suchthat∫ Sp(n,R) t(g)dg= c2 ∫ K ∫ a+ ∫ K t(k2.exp(Adk1(H)))J(H)dk1dHdk2, (13) wheredg is theHaarmeasureonSp(n,R)and J(H) = ∏ λ∈Λ+ eλ(H)−e−λ(H) = 2|Λ +|∏ λ∈Λ+ sinh(λ(H)). Thus, forall t∈Cc(Sp(n,R)/K), ∫ Sp(n,R)/K t(x)dx= c2 ∫ K ∫ a+ t(exp(Adk(H)))J(H)dkdH, (14) where dx is the invariantmeasure on Sp(n,R)/K. After identifying Sp(n,R)/K and exp(p), the Riemannianmeasureon exp(p) coincideswith the invariantmeasureonSp(n,R)/K. Thus, for all t∈Cc(exp(p)), ∫ exp(p) t(x)dvol′= c2 ∫ K ∫ a+ t(exp(Adk(H)))J(H)dkdH. (15) UsingthenotationHYofEquation(12), ∫ p˜ t(exp(Y))J(HY)∏ λ∈Λ+ 1 λ(HY) dY= c1 ∫ K ∫ a+ t(exp(Adk(H)))J(H)dkdH. (16) CombiningEquations (15)and(16),weobtain that thereexists c3 suchthat ∫ p˜ t(exp(Y))∏ λ∈Λ+ sinh(λ(HY)) λ(HY) dY= c3 ∫ exp(p) t(x)dvol′. (17) Theterm sinh(λ(H)) λ(H) canbeextendedbycontinuityona; thus, ∫ p t(exp(Y))∏ λ∈Λ+ sinh(λ(HY)) λ(HY) dY= c3 ∫ exp(p) t(x)dvol′. (18) LetdYbetheLebesguemeasurecorrespondingto themetric. Then, theexponentialapplication doesnot introduceavolumechangeat 0∈ p. SinceH0 = 0and sinh(λ(H))λ(H) −→H→0 1,wehave c3 = 1. Let logdenote the inverseof theexponentialapplication.Wehave dlog∗(vol′) dY = ∏ λ∈Λ+ sinh(λ(HY)) λ(HY) . 354