Page - 380 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 (see [10],p. 24). Let ai= eri, thensimplecalculationsshowthat: ds2(Y)= m ∑ j=1 dr2j +8∑ i<j sinh2 (ri−rj 2 ) θ2ij Asaconsequence, thevolumeelementdv(Y) iswrittenas: dv(Y)=8 m(m−1) 4 det(θ)∏ i<j sinh (|ri−rj| 2 ) m ∏ i=1 dri Thisprovestheproposition(thefactorm!2m comesfromthefact that thecorrespondencebetween Yand (r,U) isnotunique:m! correspondstoallpossiblereorderingsof r1, . . . ,rm , and2m corresponds to theorientationof thecolumnsofU). B.DerivationofEquation (19) ByEquations (16) and (18), toproveEquation (19), it is sufficient toprove that forallY∈Pm, d(Y, I) = (∑mi=1r 2 i) 1/2 if the spectraldecompositionofY isY=U†diag(er1, · · · ,erm)U,whereU is anorthogonalmatrix. Note that d(Y, I) = d(diag(er1, · · · ,erm).U, I) = d(diag(er1, · · · ,erm).U, I.U), where . is the affine transformation given by Equation (9). By Equation (10), it comes that d(Y, I)= d(diag(er1, · · · ,erm), I), and so, d(Y, I) = (∑mi=1r2i)1/2 holds using the explicit expressionEquation(8). C.TheNormalizingFactorζm(σ) Thesubjectof this section is toprove these twoclaims: (i) 0<σm<∞ forallm≥2; (ii) σ2= √ 2. To check (i), note that∏i<jsinh ( |ri−rj| 2 ) ≤ exp(C|r|) for some constantC. Thus, for σ small enough, the integral Im(σ) = ∫ Rm e −|r|σ ∏i<jsinh ( |ri−rj| 2 ) dr given in Equation (19) is finite, and consequently,σm>0. FixA>0, suchthatsinh(x2)≥ exp(x4) forallx≥A. Then: Im(σ)≥ ∫ C exp ( 1 4∑i<j (rj−ri)− |r|σ ) dr whereC is thesetof infiniteLebesguemeasures: C={r=(r1, · · · ,rm)∈Rm : ri∈ [2(i−1)A,(2i−1)A],1≤ i≤m−1,rm≥2(m−1)A} Now: 1 4∑i<j (rj−ri)= 14rm+ 1 4 (−r1+ ∑ i<j,(i,j) =(1,m) (rj−ri)) Assumem≥ 3 (the casem= 2 is easy todealwith separately). Then, onC, 14∑i<j(rj−ri)≥ 1 4rm+C ′ and |r|σ ≤ (C ′′+r2m) 1 2 σ , where C ′ and C′′ are two positive constants (not depending on r). However, forσ largeenough: 1 4∑i<j (rj−ri)− |r|σ ≥ 1 4 rm+C′− (C ′′+r2m) 1 2 σ ≥0. andso, the integral Im(σ)diverges. Thisshowsthatσm isfinite. 380