Geometric structures on the non-parametric statistical manifold

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1 UNIVERSITÀ DEGLI STUDI DI MILANO Fcoltà di Mtemtic Corso di Dottorto in Mtemtic Tesi di Dottorto Geometric structures on the non-rmetric sttisticl mnifold Alberto Cen Direttore del corso di dottorto rof. Antonio Lnteri Tutore: rof. Giovnni Pistone Anno Accdemico XIII ciclo

2 Tble of contents Summry II Anlyticl frmework. Young functions nd Orlicz Sces Orlicz sces in Informtion Geometry Regulrity 9 2. Anlytic mings between Bnch sces Exonentil function ex, Anlyticity of the Moment Generting Functionl Structures of mnifold 8 3. Exonentil mnifold Mixture mnifold Christoffel symbols nd connections Connection in vector bundle Slitting Exonentil Connection Mixture Connection Geometry of the shere in L µ Amri α-embedding F α -vector bundle An exmle in the rmetric cse Finsler structure 6 I

3 Summry Sttisticl Mnifolds re mnifolds such tht ech oint cn be identified with robbility density with resect to given mesure. The theory of Sttisticl Mnifolds, the so clled Informtion Geometry, strted with er of Ro 945 where sttisticl model ws considered s Riemnnin mnifold with the metric tensor given by the Fisher informtion mtrix. Seminl work of S.-I. Amri dded some insight to Informtion Geometry. Until quite recently the theory ws develoed only in the rmetric, commuttive setting nmely finite dimensionl mnifolds. For references see: Amri 985; Amri nd Ngok The first rigorous infinite dimensionl extension hs been develoed by Pistone nd Semi 995; Pistone nd Rogntin 999. Let X,X,µ be mesure sce nd let M µ := { : X R mesurble : > 0 µ.e., } dµ = be the set of ll µ-lmost surely strictly ositive robbility densities. In the sequel M µ will lwys refer to this set. The set M µ cn be endowed with structure of C -Bnch mnifold using the Orlicz sce bsed on n exonentilly growing function. With this structure it is clled the exonentil sttisticl mnifold. In neighborhood of ech density, M µ is modeled by the subsce B of ll the centered rndom vribles of the Orlicz sce L Φ µ ssocited to the Young function Φ = cosh, with resect to the robbility mesure µ nd with the Luxembourg norm Φ,. We review the construction of the exonentil sttisticl mnifold in Section 3.. In Section 3.2 we roose to enlrge the set suorting the mnifold structure nd we consider { } P := L µ : dµ =. For ech M µ, if E is the connected comonent of M µ contining, then E E := { q P : q } LΦ 3 µ II

4 where L Φ 3 µ is the Orlicz sce ssocited to the Young function Φ 3 x = + x log + x x with resect to the robbility mesure µ. In Theorem 37 we construct C -Bnch structure on E. In neighborhood of ech f E it is modeled by the closed subsce B f of ll the centered rndom vribles of L Φ 3 f µ. In Proosition 30 we re ble to chrcterize ll the robbility densities q with finite Kullbck-Leibler divergence with resect to, [ ] q q K q = E log <. In fct we rove tht for ech robbility density q K q < q E. In Section. we briefly introduce Young functions nd Orlicz sces. It is known tht the Orlicz sces L Φ µ nd L Φ q µ re equl for ech ir of densities nd q belonging to one-dimensionl exonentil model: ϑ = e ϑu ψϑ r where r M µ, u L Φ r µ nd ϑ δ,δ R. Using the fct tht n Orlicz sce is Bnch idel sce, in Proosition 4 we resent new roof of the equivlence between the Luxembourg norms Φ, nd Φ,q. The im of Chter 2 is the study of the regulrity of the moment generting functionl M : L Φ µ u E e u = e u dµ [0, + ] In Theorem 22 we rove tht the functionl M is nlytic on the oen unit bll of L Φ µ. For ech u in neighborhood of u 0 B 0,, M hs the ower exnsion: M u = n! E [λ,n u 0 u u 0 n ] n=0 where λ,n u 0 L n s L Φ µ ; L µ for is the n-multiliner symmetric function defined by λ,n u 0 w,...,w n = w wn e u0. nd λ,0 u 0 = e u 0/ L µ. With these multiliner ms, in Definition 9, we introduce the exonentil m ex, between the oen unit bll of the Orlicz sce L Φ µ nd the Lebesgue sce L µ for ech nd for ech density III

5 M µ. Using ex,, in Proosition 75, we re ble to rove tht the Amri α-embedding M µ L µ. is nlytic. Amri 982 nd, indeendently, Čencov 982 introduced fmily of ffine connections α with α [,] in the rmetric cse. Gibilisco nd Pistone 998 show tht the retngent nd the tngent bundle over the Exonentil Sttisticl Mnifold re the nturl domins to define, using rllel trnsort, resectively the mixture nd the exonentil connection in the non-rmetric cse. Then they define the infinite-dimensionl version of the α-connections on suitble fmily of vector bundles, the so clled α-bundle. In this infinite dimensionl contest, we evlute the Christoffel symbols of the exonentil nd mixture connection see resectively Definition 6 nd 64. In Section 4.5 we study the regulrity of the shere in the Lebesgue sces nd in Proosition 74 we find the Christoffel symbols of the nturl connection on its tngent bundle. In Proosition 78 we give the Christoffel symbols of the α-connection on the F α -bundle over M µ then, in Theorem 82, we rove tht they define the connection which is the ull-bck of the nturl connection on the shere by the Amri embedding. In Chter 5 we show tht the exonentil sttisticl mnifold dmits lso Finsler structures. We hoe tht Finsler structure cn give new insight into Informtion Geometry. IV

6 Chter Anlyticl frmework. Young functions nd Orlicz Sces Young functions re generliztion of the fmily of mings x x / with nd Orlicz sces re generliztion of Lebesgue sces L. We begin by reclling their definitions nd some mteril relevnt in the construction of the non-rmetric sttisticl mnifold. For generl reference see Ro nd Ren 99. A Young function Φ is convex function Φ : R R = R {+ } such tht: i Φ 0 = 0; ii Φ x = Φ x, x R; iii lim x Φ x = +. The cse Φ x = + for ll x x 0 > 0 is ermitted. Any Young function Φ dmits n integrl reresenttion Φ x = x 0 ϕ t dt x [0, where ϕ : [0, [0, ] is nondecresing, left continuous function such tht ϕ 0 = 0 nd ϕ x = + for x x 0 when Φ x = + for x x 0 > 0. Let ψ be the generlized inverse of ϕ, tht is the function ψ s = inf {t : ϕ t s} s [0,, then the conjugte function of Φ is the Orlicz function Ψ defined by Ψ y = y 0 ψ t dt y 0.

7 Anlyticl frmework Functions / nd b /b with,b > such tht + b = b re ir of conjugte Young functions. In the extreme cse = nd b =, the conjugte of the Young function x x is the ming { 0 if x < x + if x. The clssicl Young Inequlity xy x / + y b /b is generlized by xy Φ x + Ψ y x,y R where equlity holds if y = ϕ x or x = ψ y. Let X,X,µ be mesure sce. The Orlicz clss L Φ µ ssocited to the Young function Φ is the set defined by { } L Φ µ := u : X R, mesurble : Φ u dµ <. L is the Orlicz clss of the Young function /. However integrbility of Φ u is not enough to generlize the Lebesgue sces since L Φ µ my not be vector sce. The Orlicz sce L Φ µ ssocited to the Young function Φ is the set of ll the mesurble functions u : X R such tht Φ αu is µ-integrble for some α > 0: { } L Φ µ := u : X R, mesurble : α > 0 s. t. Φ αu dµ <. L Φ µ is convex vector sce. Inclusion L Φ µ L Φ µ esily follows by their definitions. In Lemm 5 we will see condition, the 2 -condition, sufficient for their equlity. If functions which differ only on sets of mesure zero re identified, L Φ µ is turned into Bnch sce by the Luxembourg norm defined by u Φ := inf { k > 0 : u Φ dµ k For ech u L Φ µ \ {0}, u Φ dµ.. u Φ In fct, there exists sequence k n u Φ such tht k n 0 nd u Φ dµ.2 k n 2 }.

8 . Young functions nd Orlicz Sces nd, using the Ftou Theorem, one cn see tht the Inequlity. follows by the bound.2. Let Φ be invertible when restricted to the ositive xis. If u L Φ µ \ {0} is fixed, then the ming k Φ u k dµ is strictly decresing in the intervl 0, nd the following equlity holds u Φ dµ =. u Φ In rticulr, in the cse of Φ = /, for ech u L µ \ {0} we hve: u u Φ u dµ = u Φ = dµ = u. If Φ nd Ψ re conjugte Young functions, then for ech u L Φ µ \ {0} nd v L Ψ µ \ {0} by the Young Inequlity we hve u v u v Φ + Ψ u Φ v Ψ u Φ v Ψ nd, using Eq.. fter integrting ech side of the bove inequlity, we hve uv u v dµ Φ dµ + Ψ dµ 2 u Φ v Ψ u Φ v Ψ so, finlly, we obtin the generlized Hölder Inequlity uv dµ 2 u Φ v Ψ u L Φ µ, v L Ψ µ..3 By.3 we see tht conjugcy imlies tht conjugte Orlicz sces re in dulity reltion, tht is there exists continuous biliner form L Φ µ L Ψ µ u,v uvdµ R. Hence we hve L Ψ µ L Φ µ but, in generl, L Φ µ nd L Ψ µ re not dul. If Φ nd Ψ re conjugte Young functions, then the Orlicz norm in L Φ µ is defined by { N Φ u := su uv dµ : v L Ψ µ, Luxembourg nd Orlicz norms re equivlent. 3 } Ψ v dµ.

9 Anlyticl frmework Young functions Φ nd Φ 2 re sid to be equivlent if there exist two constnts 0 < c c 2 < nd x 0 > 0 such tht Φ c x Φ 2 x Φ c 2 x x x 0. If Φ nd Φ 2 re equivlent, the Orlicz sces L Φ µ nd L Φ 2 µ re equl s sets nd hve equivlent norms s Bnch sces. Young functions re clssified ccording to their growth roerties. In the sequel we will need the following condition. A Young function Φ : R R + stisfies the 2 -condition if Φ 2x kφ x x x for some constnt k > 0. For exmle, function / stisfies the 2 -condition with Inequlity.4 holding for every rel number with constnt equl, for exmle, to 2..2 Orlicz sces in Informtion Geometry We introduce the Orlicz sces used in the non-rmetric Informtion Geometry. For i =,2,3 let Φ i : R [0, be the following Young functions Φ : x cosh x Φ 2 : x ex x x Φ 3 : x + x log + x x. Henceforth Φ, Φ 2 nd Φ 3 will lwys refer to these rticulr functions. Φ nd Φ 2 re equivlent. Φ 2 nd Φ 3 re ir of conjugte functions. Φ 3 stisfies the 2 -condition with Inequlity.4 holding for ll x R if we tke constnt equl to 4. For ech ositive density M µ, the Orlicz sce L Φ µ cn be chrcterized by the following Proosition 2. Let u be rndom vrible on X,X, µ. The moment generting function of u is the m û : R [0, + ] defined by û t := e tu dµ = E e tu i.e. it is the Llce trnsform of the distribution of the rndom vrible u with resect to the robbility mesure µ. Definition. For ech density M µ, the Crmér clss t is the set of ll the rndom vribles u on X,X, µ such tht moment generting function û with resect to the robbility mesure µ is finite in neighborhood of 0. 4

10 .2 Orlicz sces in Informtion Geometry µ-integrbility of e tu for t in neighborhood of 0 imlies tht u hs finite execttion. The subset of ll the rndom vribles with zero execttion is clled the centered Crmér clss. Proosition 2. For ech density M µ, L Φ µ coincides with the Crmér clss t. The centered Crmér clss is closed subsce. Proof. See Pistone nd Rogntin 999, Proosition 3. Definition 3. For ech density M µ, we shll denote by B the closed subsce of ll the centered rndom vribles in L Φ µ, tht is B = { } u L µ : 0 dom û, E u = 0 = { u L Φ µ : E u = 0 }. It inherits the structure of Bnch sce with the Luxembourg norm { u ] } u Φ, = inf k > 0 : E [cosh k for ech u B. Let F S be subset of the set S of ll mesurble functions on mesure sce. F, is Bnch idel sce if i it is Bnch sce; ii for ech u S nd v F, u v imlies u F ; iii the norm is monotone: for ech u,v F, u v imlies u v. Orlicz sces re Bnch idel sces. From the min result of Pistone nd Semi 995 follows the equlity s sets nd Bnch sces between L Φ µ nd L Φ q µ for ech ir of densities nd q belonging to one-dimensionl exonentil model: ϑ = e ϑu ψϑ r where r M µ, u L Φ r µ nd ϑ δ,δ R. Pistone nd Rogntin 999, Proosition 5 give direct roof of the equlity only s sets. In the following roosition we use the fct tht Orlicz sces re Bnch idel sce to rove the equivlence between the norms Φ, nd Φ,q. 5

11 Anlyticl frmework Proosition 4. Let nd q be densities connected by one-dimensionl exonentil model, then the identity m id : L Φ µ, Φ, L Φ q µ, Φ,q is n homeomorhism. Proof. We know tht L Φ µ = L Φ q µ so the identity m id is defined. We hve to rove its continuity. Let {u n } L Φ µ be sequence converging in norm Φ, to 0. We suose it doesn t converge in norm Φ,q. In rticulr, if necessry considering subsequence, we suose u n Φ,q > ε..5 By the convergence in norm Φ, there is subsequence {u nk } such tht Since u nk Φ, < 2 k. ku nk Φ, k= k= k 2 k < nd L Φ µ, Φ, is comlete, series k unk converges in norm nd let r L Φ µ be such tht k u nk = r. Clim. Let {x n } n N nd {y n } n N be two sequences in Bnch idel sce such tht x n y n for ech n N. If x n x nd y n y in norm, then x y see Kntorovitch nd Akilov 980,. 42. For ech m N, let r m L Φ µ be the rtil sum r m = m k= k u n k. Sequence r m converges to r in norm Φ,. For ech k > 0, k u nk r m with m k. Fix k > 0, considering the constnt sequence {x m := u nk } m k nd the sequence {r m } m k, by the clim we hve u nk < r k. By the monotony of the norm Φ,q we hve the bound u nk Φ,q < k r Φ,q 0 for k in contrst with the condition.5. Hence the identity m id is continuous nd, by symmetry, it is n homeomorhism. 6

12 .2 Orlicz sces in Informtion Geometry Now we tret the Orlicz sce L Φ 3. Lemm 5. Let X,X,µ be finite mesure sce. If Young function Φ stisfies the 2 -condition then the Orlicz clss L Φ µ coincides with the Orlicz sce L Φ µ. Proof. We hve to rove only the inclusion L Φ µ L Φ µ. Let u 0 be n element of the Orlicz sce L Φ µ with Luxembourg norm u Φ. There exists n N such tht u Φ 2 n. Since Φ is even, incresing for ositive rel vlues nd it stisfies Inequlity.4 for x x 0 0, we hve u Φ u dµ = Φ u Φ dµ u Φ Φ 2 n u dµ u Φ u k n Φ dµ + µ X Φ 2 n x 0 u Φ k n + µ X Φ 2 n x 0 <. Hence Φ u is µ-integrble nd u belongs to the Orlicz clss L Φ µ. If the mesure sce is not finite but the Young function Φ stisfies the 2 - condition with Inequlity.4 holding for ll the rel numbers, then Lemm 5 is lso true. In fct, in this cse let u L Φ µ be mesurble function such tht Φ αu is µ-integrble for α > 0, if we tke m N such tht 2 m α, since Φ is even, incresing for ositive rel vlues nd it stisfies Inequlity.4 for every rel vlue we hve Φ u Φ 2 m αu k m Φ αu. Hence Φ αu dµ < imlies tht Φ u is µ-integrble. Proosition 6. Let X,X, µ be robbility sce. A rndom vrible u belongs to L Φ 3 µ if nd only if + u log + u is µ-integrble. Proof. Let u be rndom vrible. Since Φ 3 stisfies the 2 -condition, by Lemm 5 we hve u L Φ 3 µ = L Φ 3 µ E [ + u log + u u ] <. Since + x log + x x > x for x > x 0 > 0, we hve { E u < nd E [Φ 3 u] < E [ + u log + u ] < E [ + u log + u ] <..6 7

13 Anlyticl frmework Remrk. Equivlence in line.6 shows in rticulr tht if u L Φ 3 µ then u L µ. A bsic result in the theory of the Lebesgue sce is, in the cse of finite mesure sce Ω,Σ,m, the chin of inclusions L m L r m L s m L m where s < r. Symbol mrks the continuity of the inclusion. In fct we hve the bound s k r where k = k r,s = {m Ω r s rs if r < m Ω s if r =. In our cse we cn comlete with the following chin of inclusion: L µ L Φ µ L r µ L s µ L Φ 3 µ L µ. Continuity follows from the existence of constnts k, k 2, k 3 nd k 4 such tht k Φ3 k 2 s k 2 r k 3 Φ k 4. Such inequlities deend on the different growth t of the Young functions involved. Definition 7. For ech density M µ, we shll denote by B or by L log L 0 the so clled x log x-clss, tht is the subsce of ll the centered rndom vribles in L Φ 3 µ: B = L log L 0 := { v L Φ 3 µ : E v = 0 }. Proosition 8. For ech M µ we hve the following two sttements: i All the elements u B re identified with n element u of the dul sce B of B by the formul: u v = E uv, with v B. In rticulr, B is identified with roer subset of B nd the injection of B into B is continuous; we write B B. ii All the elements u B re identified with n element u of the dul sce B of B by the formul u u = E u u, with u B. This identifiction is onto, tht is B is identified with B ; we write: B B. Proof. See Pistone nd Rogntin 999, Proosition 8. 8

14 Chter 2 Regulrity Before reclling the construction of the Exonentil Sttisticl Mnifold in the following Chter we define the exonentil ming nd we study the regulrity of the moment generting functionl. Definition 9. The moment generting functionl M : L Φ µ [0, + ] is defined by M u := E e u. Pistone nd Semi 995 rove tht M is Gâteux-differentible in the interior of its roer domin. This result comes from smoothness of the Llce trnsform û for ech u domm. Then they rove tht M is infinitely Fréchet-differentible on the oen unit bll B 0, L Φ µ. Pistone 200 suggests tht the m M is lso nlytic B 0,. Before roving this, we recll some bsic fcts regrding nlytic functions between rel Bnch sces t introductory level, see Prodi nd Ambrosetti 973 nd Umeier 985, Sections I.-I.2; for series in Bnch sces see Kdets nd Kdets Anlytic mings between Bnch sces Let E nd F be Bnch sces. We denote by L E; F the Bnch sce of the continuous liner ms of E into F, by L n E; F the Bnch sce of the continuous n-multiliner ms of E n into F. The subset of those which re symmetric is denoted by L n s E; F. In the cse F = R these sces re denoted by L E, L n E nd L n s E. 9

15 2 Regulrity Definition 0. A function ˆλ : E F between Bnch sces is clled continuous n-homogeneous olynomil if there exists λ L n s E; F such tht ˆλ x = λx n x E. Let P n E; F be the vector sce of ll continuous n-homogeneous olynomils from E to F. Let P 0 E; F := F. The n-multiliner symmetric function λ is uniquely determined by ˆλ in fct, D nˆλ = n!λ nd it is clled the olr form of ˆλ. We endow P n E; F with the norm: ˆλ x ˆλ := su F P n x 0 x n = su ˆλ x. E x E = F Clerly ˆλ λ P n L n. There is n inequlity in the oosite direction: } Definition. Let {ˆλn ˆλ n N P n λ L n nn n! ˆλ P n. 2. be sequence with ˆλ n P n E; F, then ˆλ n 2.2 n=0 is clled ower series from E to F. The rdius of convergence ˆρ of 2.2 is defined s: { } P ˆρ := su r [0, : ˆλ n r n <. 2.3 n The rdius of restricted convergence ρ of 2.2 is defined s { } ρ := su r [0, : λ n L n r n <. n=0 n=0 For the Cuchy-Hdmrd formul, n = lim su ˆλ n ˆρ n P n nd ρ By 2. nd reclling tht lim n n n n! /n = e, we hve ˆρ e ρ ˆρ. 0 = lim su λ n n L n. n

16 2. Anlytic mings between Bnch sces P The inequlity ˆλ n x ˆλ n x n F n E imlies tht the series ˆλ n=0 n x converges bsolutely nd uniformly in the closed bll B 0,r for ech r < ˆρ, moreover Umeier, 985, Proosition.4 { } ˆρ = su r : ˆλ n x conv. uniformly for x E r. 2.4 n=0 Similrly, series n=0 λ n x,...,x n converges uniformly for every sequence {x m } m such tht su { x m E } r with fixed r < ρ nd { } ρ = su r : λ n x,...,x n conv. unif. {x m } m s.t. su x m E r. n=0 Definition 2. Let E nd F be Bnch sces nd U E be oen. A ming f : U F is clled nlytic if for ech x 0 U there exists convergent ower series ˆλn with ositive rdius of convergence such tht, for ech x in neighborhood of x 0, f x = ˆλ n x x n=0 Series ˆλ n=0 n x is n nlytic function in the bll B 0,ρ of restricted convergence. An nlytic function f is infinitely often differentible. If f hs the ower exnsion 2.5, then D k f x 0 = k!λ k nd D k f hs the following ower exnsion bout x 0 D k k + n! f x = λ k+n x x0 n,. 2.6 n! n=0 Series on the right-hnd side of Equtions 2.5 nd 2.6 hve the sme rdius of restricted convergence. Like in the rel cse, there re the following theorems for the reresenttion in ower series of smooth functions nd for the comosition of nlytic functions Prodi nd Ambrosetti, 973, Theorem 0.5 nd.. Theorem 3. Let E nd F be Bnch sces nd U E be oen. Let f : U F be smooth function. If there exists constnt M such tht for ll x B x 0,r U D n f x L n Mn! r n then f is reresented by the Tylor series: f x = n=0 n! Dn f x 0 x x 0 n

17 2 Regulrity nd f is nlytic in B x 0,r. Theorem 4. Let E, F nd G be Bnch sces, U E nd V F be oen. If f : U F nd g : V G re nlytic functions such tht f U V then g f is lso nlytic. 2.2 Exonentil function ex, We shll define n nlytic function between the oen unit bll B 0, of the Orlicz sce L Φ µ nd the Lebesgue sce L µ. Lemm 5. In the Orlicz sce L Φ µ with the Luxembourg norm, u Φ E cosh u 2. Proof. As we observed in Section., since Φ is strictly incresing nd continuous on [0,, the ming 0, k E Φ u/k R is strictly decresing for ech 0 u L Φ µ nd u Φ E Φ u E cosh u 2. Lemm 6. For ech, n N nd u B 0,, let λ,n u be defined by: λ,n u : L Φ µ L Φ µ L µ w,...,w n w wn Then λ,n u re continuous, symmetric, n-multiliner mings. Proof. Put r = n u Φ. For ech v,...,v n L Φ µ with v i Φ =, we hve n u + r v i u Φ + rn = Φ i= so u + r n i= v i B 0,. By the inequlity x < e x for ech x R nd by Lemm 5, we hve r v r v n e u r v dµ = r v n e u dµ [ E e u+r ] n i= v i ] n 2E [cosh u + r v i 4 2 i= eu/

18 2.2 Exonentil function ex, hence v vn eu/ dµ 4 r n = 4n n u Φ n = C n u <. 2.7 For rbitrry w,...,w n L Φ µ \ {0} if v i = w i / w i Φ, then w wn eu/ dµ = w Φ w n Φ E [ v vn eu/ ]. 2.8 By Equtions 2.8 nd 2.7 we hve hence λ,n u L n s w λ,n u w,...,w n = wn eu/ dµ 2.9 C n u / w Φ w n Φ L Φ µ ; L µ. If u B 0,, then e u/ L µ. In fct, since u Φ <, e u/ [ dµ E ] [ ] e u u 2E [cosh u ] 2E cosh 4. u Φ Definition 7. For ech nd u B 0,, let ˆλ,0 u := e u/ nd let ˆλ,n u P n L Φ µ,l µ with n be the n-homogeneous olynomil determined by the olr form λ,n u defined in Lemm 6. Proosition 8. Let M µ nd. Series A v = n=0 v n n! is ower series from L Φ µ to L µ with rdius of convergence ˆρ. Proof. By Lemm 6 A v = n=0 v n = n! n! λ,n 0 v n n=0 3

19 2 Regulrity nd A = n! ˆλ,n 0 is ower series from L Φ µ to L µ. Let v L Φ µ be such tht v Φ =. By the inequlity n! hve n! ˆλ,n 0 v = v n dµ n! E [ e v ] 2E [cosh v ] 4 nd ˆλ n!,n 0 4 /. Hence /ˆρ lim n 4 /n = nd ˆρ. Pn v n e v /, we 2.0 Definition 9. For ech nd for ech density M µ, the exonentil function ex, between the oen unit bll of the Orlicz sce L Φ µ nd the Lebesgue sce L µ is defined by ex, : B 0, L µ v v n. n! Let v B 0, L Φ µ. We know tht, ointwise, v k k=0 k! = e v/. Since µ is finite mesure, n v k k=0 k! e v/ lso in µ-robbility. Proosition 8 shows tht it converges to ex, v in norm so it is men convergent to ex, v in L µ. Since convergence in men imlies convergence in mesure see Semi 986, n k k=0 ex, v in µ-robbility nd, by uniqueness, k! v n=0 ex, v = e v. Proosition 20. Exonentil function ex, stisfies the following roerties: i ex, 0 = ; ii for ech u,v B 0, such tht u + v B 0, ex, u + v = ex, u ex, v ; iii for ech u B 0,, ex, u hs n inverse ex, u in L µ nd ex, u = ex, u. 4

20 2.3 Anlyticity of the Moment Generting Functionl Proof. i ex, 0 = λ,0 0 = e 0 =. ii Since in Bnch sce bsolute convergence imlies unconditionl convergence converse is true in R n, but not in the generl cse, with rerrngement of terms we hve: ex, u + v = = n=0 k=0 n! n m=0 k! u n u n m v m m k m=0 iii It comes from i nd ii utting v = u. v m = ex, u ex m!, v. Theorem 2. Let M µ nd. Ming ex, is n nlytic function. In neighborhood of ech u 0 B 0, it cn be exnded in the Tylor series u n u 0 ex, u 0 + u = e. 2. n! Proof. The uer bound of λ,n 0 L n coming from 2.9 suffices to demonstrte nlyticy of ex, only in neighborhood of 0. In fct, n! λ,n 0 = su λ n!,n 0 w,...,w n C n 0 = 4 n n L n w Φ w n Φ n! n! w i Φ 0 nd, since lim n λ n!,n 0 /n L n limn 4 / n n /n n! = e, for the Cuchy Hdmrd formul, we find tht the rdius of restricted convergence ρ is not lower thn /e. Let u 0 B 0,. Reeting revious rgument for the n-homogeneous olynomil ˆλ n!,n u 0, we find tht the ower series ˆλ n=0 n!,n u 0 hs rdius of restricted convergence ρ u 0 Φ /e > 0. Hence in neighborhood of u0, ming u n=0 ˆλ n!,n u 0 u is nlytic nd n! ˆλ u,n u 0 u = n! n=0 n=0 n=0 n e u 0 = e u 0 +u = ex, u 0 + u. 2.3 Anlyticity of the Moment Generting Functionl We re now ble to imrove the results of Pistone nd Semi 995 bout the regulrity of the Moment Generting Functionl. We first note tht M, restricted to the oen unit bll B 0,, coincides with E ex,. 5

21 2 Regulrity Theorem 22. The moment generting functionl M stisfies the following roerties: i M 0 = otherwise, for ech u 0, M u > ; ii it is convex nd lower semicontinuous, its roer domin domm = { u L Φ µ : M u < } is convex set which contins the oen unit bll B0, L Φ µ; iii it is infinitely Gâteux-differentible in the interior of its roer domin, the nth-derivtive t u domm in the direction v L Φ µ is d n dt M u + tv n = E v n e u ; t=0 iv it is bounded, infinitely Fréchet-differentible nd nlytic on the oen unit bll of L Φ µ, the nth-derivtive t u B0, evluted in v,...,v n L Φ µ L Φ µ is In rticulr, DM 0 = E. D n M uv,...,v n = E v v n e u. Proof. i ii See Pistone nd Semi 995. iii If u enough domm, for every v L Φ µ, u + tv M u + tv = e tv e u dµ = M ue q e tv domm for t smll where q = e u /M u. E q e tv is the Llce trnsform of v with resect to the robbility density q, it is nlytic in the interior of its roer domin see Pistone nd Semi 995 nd E q e tv E q v k = t k k! hence nd k=0 d n dt n M u + tv = M u k=n E q v k k n! tk n d n dt M u + tv n = M u E q v n = E v n e u. t=0 6

22 2.3 Anlyticity of the Moment Generting Functionl iv For ech u B 0, nd n N, we hve E ˆλ,n u P n L Φ µ ; R nd n=0 E n! ˆλ,n u is ower series from L Φ µ to R with ositive rdius of convergence. In neighborhood of ech u 0 B 0,, by 2. we hve M u = E ex, u = n=0 n! λ,n u 0 u u 0 n dµ. Integrting term by term we obtin the following exnsion in ower series bout u 0 : M u = = n=0 n=0 n! E ˆλ,n u 0 u u 0 n! E u u 0 n e u 0. Hence M is n nlytic function. Its nth-derivtive t u in the directions v,...,v n L Φ µ L Φ µ is D n M u v,...,v n = E λ,n u v,...,v n = E v v n e u. Definition 23. The cumulnt generting functionl K : B [0, + ] is defined by K u := log M u. Corollry 24 of Theorem 22. For ech density M µ, the cumulnt generting functionl K : B B 0, [0, is n nlytic function. Proof. It is comosition of nlytic functions. 7

23 Chter 3 Structures of mnifold 3. Exonentil mnifold First we review the construction, due to Pistone nd Semi 995, of the structure of mnifold over M µ. We refer to the resenttion of infinite dimensionl mnifold of Lng 995. For ech M µ let V be the oen unit bll in the Orlicz sce B, tht is } V := {u B : u < Φ,. It is subset of the roer domin of the cumulnt generting functionl K nd we cn define the ming e : V M µ u e u Ku This function is one to one. In fct if e u = e u 2, then u K u = u 2 K u 2 nd u u 2 is constnt. Since u,u 2 B this constnt hs to be 0. The rnge of e is denoted by U. The inverse of e on U is the function s : U q log q E [ log q ] V. Ech ir U,s for M µ is chrt. The centered log-likelihood s is the coordinte function. If U U 2 for ir of densities, 2 M µ, then the trnsition ming s 2 e is the ffine function s 2 e : s U U 2 s 2 U U 2 [ u u + log E 2 u + log ].

24 3.2 Mixture mnifold The derivtive of the overl m s 2 e is B u u E 2 u B 2 which is toliner isomorhism between B nd B 2. Definition 25. The sequence { n } n N in M µ is e-convergent exonentilly convergent } to if { { n } n N tends to in µ-mesure s n nd moreover the sequence nd re eventully bounded in ech L α µ, α >, tht is { n n N α > : n }n N [ α ] n lim su E n [ α ] <, lim su E <. n n We reort the min Theorem of Pistone nd Semi 995 Theorem 26. The collection of irs {U,s : M µ } is n ffine C tls on M µ. The induced toology is equivlent to the e-convergence. Definition 27. The exonentil sttisticl mnifold is the mnifold defined by the tls in Theorem 26. Definition 28. For every density M µ, the mximl exonentil model t is defined to be the fmily of densities { } E := e u Ku : u dom K M µ. Pistone nd Semi 995, Theorem 4. rove tht E is the connected comonent of the mnifold M µ contining. 3.2 Mixture mnifold Let M µ be robbility density. For ech u V nd q = e u Ku, the derivtive of K t u, DK u B, is the liner ming [ ] q DK u v = E v, v B nd DK u is identified to q/ B ccording to Proosition 8 see Pistone nd Rogntin 999, Proosition 6-d. Ming U q q/ B cnnot be chrt becuse the vlues re bounded below by. 9

25 3 Structures of mnifold We enlrge the mximl sttisticl model M µ not ssuming the condition > 0 ny more. Let us consider the set of ll the robbility densities reltive to the mesure µ P µ := { L µ : 0, } dµ = nd the set P := { L µ : } dµ =. Observe tht M µ P µ P. For ech q P there exists n element q P µ defined by q := q. q dµ by For ech robbility density P µ, let us introduce the subset U of P defined U := { q P : q } LΦ 3 µ. Then consider the following m η defined on U η : U B This ming is bijective nd its inverse is: q q. B u u + U. The collection of sets { U } Pµ is covering of P. In fct, for ech q P we hve q U q. Let us chrcterize the elements of U Pµ : they re ll the robbility densities with definite Kullbck-Leibler divergence with resect to. Definition 29. Let M µ nd q P µ be given. If q/ log q/ is µ integrble, then the Kullbck-Leibler divergence or reltive entroy of q with resect to is the number: [ ] q q K q = E log. Proosition 30. Let M µ be given. For ech q P, the reltive entroy K q of the robbility density q with resect to is definite if nd only if q U : [ ] q q K q = E log < q U. 20

26 3.2 Mixture mnifold Proof. Let q P. We ssume tht K q <. By the inequlity Φ 3 x + x log x for x > 0 we hve ] [ ] q q E [Φ 3 q dµ = E Φ 3 [ ] q q + E log = + K q <. Hence q LΦ 3 µ nd, by definition, q U. Conversely, let q U, tht is q LΦ 3 µ. Since L Φ 3 µ is liner, q dµ q LΦ 3 µ nd, by Proosition 6, + q log + q is µ integrble. From the inequlity x log + x + x log + x, x > 0 where log + x := mx {0, log x}, we hve: [ ] [ ] q E q q q log E log+ + [ E + q log + q ] + < hence K q is defined. Corollry 3. The reltive entroy K q of robbility density q P µ with resect to M µ is defined if nd only if log L q µ. Proof. If q P µ, then q = q nd we hve [ ] q q K q < E log < [ E q log q ] q <. Proosition 32. Let M µ be given, then U U. Proof. If q U, there exists u V B such tht q = e u Ku. Since L Φ q µ = L Φ µ the rndom vrible u is q µ-integrble nd we hve ] q E q [log = E q u K u <. 3. From the Corollry 3 follows tht q U. 2

27 3 Structures of mnifold Using results of Section 2.3 nd Eqution 3., we cn conclude tht for ech M µ the function where q = e u Ku is smooth. In fct V u K q R K q = E q u K u = E ue u E e u K u = DM u u M u K u = DK u u K u. Proosition 33. Let nd 2 be two densities in the sme connected comonent E M µ for some density M µ. Then the ming is toliner isomorhism. U 2 : L Φ 3 µ L Φ 3 2 µ u u 2 Proof. Let u L Φ 3 µ. The Orlicz norm N Φ3, 2 of u / 2 is, if it exists, the number N Φ3, 2 u 2 { = su = su u 2 } v 2dµ : v L Φ 2 2 µ,e 2 [Φ 2 v] }. { E uv : v L Φ 2 2 µ, v Φ2, 2 Since Φ nd Φ 2 re equivlent norms nd the Bnch sces L Φ µ nd L Φ 2 µ coincide, lso the Bnch sces L Φ 2 µ nd L Φ 2 2 µ coincide nd there exists constnt c = c, 2 > 0 such tht c v Φ2, v Φ2, 2 for ech v L Φ 2 µ = L Φ 2 2 µ. 22

28 3.2 Mixture mnifold If v Φ2, 2 then cv Φ2,. We hve N Φ3, 2 u { } = su E uv : v L Φ 2 2 µ, v Φ2 2 2 = } {ce c su uv : v L Φ 2 2 µ, v Φ2, 2 } {E c su u cv : v L Φ 2 2 µ, cv Φ2, = } {E c su uv : v L Φ 2 µ, v Φ2, = c N Φ 3, u <. Since N Φ3, 2 u / 2 is bounded, u / 2 is n element of L Φ 3 2 µ nd, since N Φ3, 2 u c N Φ3, u, 2 the liner m U 2 is continuous. Simmetriclly one cn see tht the inverse is continuous liner m. L Φ 3 2 µ w U 2 w = w 2 L Φ 3 µ Corollry 34. Let nd 2 be two ositive densities in the sme connected comonent E M µ for some density M µ. Then i 2 L Φ 3 2 µ; ii the liner m P m 2 : B B 2 is toliner isomorhism. B u P m 2 u = u 2 B 2 Proof. i We hve U 2 = 2 L Φ 3 2 µ. ii P m 2 is the restriction of U 2 to the x log x clss of L Φ 3 µ. If u B then E 2 u / 2 = E u = 0 nd u / 2 B 2. Proosition 35. Let nd 2 be ir of ositive densities in the sme connected comonent E M µ for some density M µ, then U = U 2. 23

29 3 Structures of mnifold Proof. Let, 2 E. First we note tht, 2 U U 2. In fct, by Corollry 34 we hve 2 L Φ 3 2 µ nd we conclude tht U 2. Similrly, we see tht 2 U. Every q U cn be written s u + where u = η q B nd we hve q = u + = P m 2 2, 2 u + L Φ 3 2 µ. 2 2 Hence q U 2 nd U U 2. In the sme wy we rove the oosite inclusion. For ech ir, 2 E we cn define the overl m η 2 η : B B 2 u u The function η 2 η is C -ffine m since it cn be written s the sum of the continuous liner m P m, 2 nd the constnt 2 B 2 so it C -ffine m. Definition 36. Let M µ be fixed. E is the subset of P defined by { E = q P : q } LΦ 3 µ E = U α for ech α E. E hs the structure of mnifold modeled on the Bnch sce B. Theorem 37. Let M µ be given. The collection of chrts is n ffine C -tls on E. { U α,η α : α E } Proof. The collection of sets { U α : α E } covers E. For ech ir, 2 E the set η U U 2 = E = B is clerly oen in B nd we hve just observed tht the trnsition ming η 2 η is n C -ffine function. We conclude with locl Pythgoren-tye reltion. Let M µ be given nd let s : U V nd η : U B be chrts resectively in E nd E. Let q U, u = s q nd 0 r U be given. Consider the dulity [ ] r η r,s q = E [η r s q] = E u = E r u. 24

30 3.2 Mixture mnifold As we hve u = log [ q E log ] q = log q + K q, ] q E r u = E r [log + K q q ] r = E r [log + log + K q r = K r q + K r + K q. In rticulr, η r,s q = 0 imlies the reltion K r q = K r + K q. 25

31 Chter 4 Christoffel symbols nd connections 4. Connection in vector bundle Let X be mnifold nd π : E X be vector bundle resectively modeled on the Bnch sces F nd E. Let us denote E x the fiber π x over x X. Let π E : T E E be the tngent bundle of E. We ssume tht mnifolds re objects of the ctegory Mn n of ll C n -mnifolds with n so ming of clss C n between them will be denoted morhism. Note tht if X is C n -mnifold, then the tngent bundle π : T X X lies in Mn n. We refer to Klingenberg 978 nd Lng 995. X hs n tls such tht over ech chrt U,ϕ there is triviliztion which determines locl reresenttion of the bundle, i.e. chrt π U,Φ of E such tht the following digrm is commuttive. Φ π U ϕ U E π r U ϕ ϕ U Exmle 38 The tngent bundle over the mximl exonentil model. By Definition 28, for every density f M µ the mximl exonentil model t f is the fmily of densities E f := { } e u K f u f : u dom K f M µ. 26

32 4. Connection in vector bundle Given the C -tls {U,s : E f} of E f, the tngent bundle T E f = q Ef hs triviliziting tls {T U,T s : E f} such tht the digrm B q T U π T s V B r U s is commuttive. For ll w B q T U = g U B g, we hve: V T s w = s q,w E w. Exmle 39 The re-tngent bundle over E f. Given the C -tls of E f, {U,s } Ef, the re-tngent bundle over E f is defined by T E f := q Ef B q. It hs triviliziting tls { T U, T s : E f} such tht the digrm T U π T s V B r U s is commuttive. For ll w B q T U = g U B g, we hve: T s w = s q,pq m w = w q V where P m q : B q B is the toliner isomorhism defined in Corollry 34. Definition 40. A connection in the vector bundle π : E X is vector bundle morhism H : π E π over π i.e. H is morhism mking the digrm T E π E E H E π π X 27

33 4 Christoffel symbols nd connections commuttive such tht for ech locl reresenttion induced by locl chrt U,ϕ of X with ϕ U = V F the locl reresenttive H V := Φ H T Φ T π U T Φ H π U Φ V E F E H V V E is given by: H V : x,ξ,y,η x,η + Γ V x y,ξ 4. where Γ V : V L 2 F,E; E is morhism. Mings Γ V re the Christoffel symbols of the connection. A connection in the tngent bundle π X : T X X is clled n ffine connection. If X nd E re objects of Mn n with n, we shll ssume tht H nd Γ V re C n -morhisms. In the cse of n ffine connection, E = T X is C n -mnifold nd we shll ssume tht n 2 nd tht H nd Γ V re C n 2 -morhisms. Let us comute the trnsformtion rule for Christoffel symbols under chnge of coordintes. If U,ϕ nd U 2,ϕ 2 is ir of chrts of X such tht U := U = U 2 nd if V = ϕ U nd V 2 = ϕ 2 U, then f := ϕ 2 ϕ : V V 2 is n isomorhism. Rising to the uer level, there re two chrts π U,Φ nd π U,Φ 2 of E nd there is the isomorhism F = Φ 2 Φ = f,l such tht F : V E x,ξ f x,l x ξ V 2 E where L : V Aut E is morhism Aut E L E; E is the oen subset of ll the toliner isomorhisms. Note tht DL : V L 2 F,E; E. When we lift F to the tngent bundles we obtin T F : V E F E V 2 E F E such tht T F : x,ξ,y,η f x,l x ξ,df x y,dl x y,ξ + L x η. 28

34 4. Connection in vector bundle We my summrize with the following digrm. V E F E T F V 2 E F E 4.2 r,2 r,2 V E V 2 E Φ Φ 2 π U r V ϕ F =f,l U f π ϕ 2 The locl reresenttions H V nd H V2 of H re relted by the reltion: H V = F H V2 T F nd T F V E F E H V T Φ T π U T Φ 2 H V 2 E F E HV2 V E Φ π U Φ 2 V 2 E H V = F H V2 T F η + Γ V x y,ξ = [ ] L x DL x y,ξ + L x η + Γ V2 f x Df x y,l x ξ. Hence under chnge of vribles the Christoffel symbols stisfy the following identity: Γ V2 f x Df x y,l x ξ = DL x y,ξ + L x Γ V x y,ξ 4.3 for ech x,ξ,y,η V E F E. Let X E X denote the C n X-module of ll the sections of clss C n of the bundle π, tht is the morhism ξ : X E such tht πξ = id X. Let X X without subscrit denote the set of ll C n -vector fields of X. 29 V 2 r F

35 4 Christoffel symbols nd connections Definition 4. Given connection H in vector bundle π : E X, the covrint derivtive of section ξ X E X is the morhism ξ : T X E defined by ξ := H T ξ. In locl chrt U,ϕ of X, section ξ X E X is reresented by morhism of V = ϕ U into V E which hs two comonent id V,ξ V See the following digrm ϕ U ξ π U Φ V id V,ξ V V E The morhism ξ V : V E is clled the rincil rt of the locl reresenttion of the section. Sometimes, to simlify the nottion we gree tht ξ = ξ V. In coordintes, T ξ : T U T π U is given by the locl reresenttion: V F x,y x,ξ V x,y,dξ V x y V E F E nd, by Eq. 4., for ech x,y ϕ U F the covrint derivtive of ξ is given by the following formul: ξ x,y = x,dξ V x y + Γ V x y,ξ V x. 4.4 Definition 42. Given vector bundle π : E X, covrint derivtive on the vector bundle is R-liner m stisfying the following conditions: : X X X E X X E X v,ξ v ξ. it is C n X-liner in the first vrible, tht is, for ech f C n X, fv ξ = f v ξ 2. it is derivtion on C n X,X E X in the second vrible, tht is, for ech f,ξ C n X X E X v fξ = v f ξ + f v ξ where v f is the Lie derivtive of f long v, v f := v f. From connection H defined by gluing together set of Christoffel symbols s in Definition 40 we lwys obtin covrint derivtive ccording to Definition

36 4. Connection in vector bundle Definition 43. Given connection H in vector bundle π : E X, covrint derivtive : X X X E X X E X is defined by setting: v ξ := ξ v where ξ is the covrint derivtive of the section ξ s in Definition 4. Definition 43 is consistent. In fct, by Eq. 4.4 we hve the locl formul v ξ V x = Dξ V x v V x + Γ V x v V x,ξ V x 4.5 nd, with n esy clcultion, we cn check tht the two conditions of Def. 42 re stisfied. In finite dimension, the ssocition of covrint derivtive to connection is bijective, in our more generl cse this is not lwys true see Lng 995, VIII-2. Definition 44. A bundle-connection ir E, is ir comosed of vector bundle π : E X nd covrint derivtive on E. Connection on mnifolds in R n For heling geometric understnding of the Christoffel symbols it is useful to look t the cse of mnifold embedded in R n. We refer to Boothby 975, VII 2. Let M R n be submnifold of dim m nd ξ X M be vector field to M. If σ : ε,ε M is C -curve, then usul differentition in R n induces n intrinsic on M derivtive of ξ long it Boothby, 975, Definition 2.2. Definition 45. The covrint derivtive Dξ dt the orthogonl rojection on T σt M of ξ t. of the tngent vector field ξ long σ is Let U,ϕ be locl chrt with ϕ U = V R m. We denote the locl coordintes on M by u,...,u m nd the coordintes on R n by x,...,x n. Let ϕ u = g u,...,g n u be defined by the n regulr functions g α : V R. A coordinte frme consists of the set of vectors fields F i X U defined by: F i u = Dϕ u u i = n g α u i α= u x α i =,...,m. They sn T q M R n where q = ϕ u nd, since ξ q T q M, there exist m regulr functions ξ i : V R such tht the locl reresenttion ξ = ξ ϕ cn be written s liner combintion: ξ u = m ξ i u F i u. 4.6 i= 3

37 4 Christoffel symbols nd connections Note tht ξ u,...,ξ m u is the rincil rt of the locl reresenttion of ξ: π U ξ T U T ϕ V id V,ξ,...,ξ m V R m Put ξ t = ξ σ t nd u t = ϕ σ t, then ξ t = m ξ i u t F i u t i= nd ξ = m Dξ i u u F i u + D 2 ϕ u ξ u, u 4.7 i= where Dξ i u u = m j= ξ i u j u u j nd D 2 ϕ u ξ, u = n m α= i,j= 2 g α u i u j ξ i u u j u x α. Alying rojection Π q : R n T q M with q = ϕ u, we find m Dξ dt = Dξ i u u F i u + i= n m 2 g α u i u j ξ i u u j Π q. 4.8 u x α α= i,j= There exist mn regulr functions k α : V R such tht: Π q = x α m k α u F k u k= α =,...,n. We my now introduce the clssicl Christoffel symbols: Γ k ij u := k= n 2 g α u i u j k α u u α= nd Eq. 4.8 become: m [ Dξ dt = Dξ k u u + i,j,k m ] m Γ k ij u ξ i u u j F k u. 4.9 i,j= 32

38 4. Connection in vector bundle In rticulr, if ξ = F i nd u t is defined by u l = const. for l j nd u j = t, then DF i dt = m Γ k ijf k. k= Hence Γ k ij is the kth comonent reltive to the coordinte frme of the rojection of the rte of chnge of F i long coordinte curve. The covrint derivtive Dξ is the intrinsic derivtive of ξ, tht is s seen from dt the viewoint of M. Biliner mings Γ u L 2 T q M; T q M Γ u v,w = m i,j,k= Γ k i,j u v i w j F k u re understood to be the term in Eq. 4.9 which tkes into ccount the motion long the curve of the tngent sces. A covrint derivtive long curve define covrint derivtive : X M X M X M s we my ssume η ξ q = η0 ξ q := Dξ dt where in the right-hnd side the covrint derivtive is long n rbitrry curve such tht σ t 0 = q nd σ t 0 = η t 0 = η 0 Boothby, 975, Theorem 2.. Finlly, in finite dimension, formul 4.5 become the clssicl one Prllel trnsort η ξ V = [ m η ξ k + k= t0 ] m Γ k ijξ i η j F k. Let π : E X be vector bundle nd α : J X be C -curve. A lift γ of α to E is C -curve γ : J E s.t. πγ = α. The set Lift E α of ll lifts of α to E is R-vector sce. Following roosition is stted by Lng 995, Theorem VIII-3. in the rticulr cse of n ffine connection determined by sry on X. Proosition 46. Given covrint derivtive : X X X E X X E X ssocited to connection, there exists unique liner m α : Lift E α Lift E α which in chrt U,ϕ hs locl reresenttion: i,j= α γ V t = γ V t + Γ V α V t α V t,γ V t

39 4 Christoffel symbols nd connections Proof. Lng s roof cn be dted using the trnsformtion rule 4.3 for Christoffel symbols to show tht Eq. 4.0 trnsforms in the roer wy under chnge of chrts. To be more recise: the locl reresenttion α γ V is ming J V E nd in Eq. 4.0 we re confounding it with the second comonent, its rincil rt. Likewise the cse of submnifold of R n, by Proosition 46 one my introduce n intrinsic derivtive of sections long curves. If ξ X E X is section nd γ t := ξ α t for t J, then Dξ := dt α γ. Note tht if η X X is vector field such tht α t 0 = η α t 0 for some t 0 J, then α γ t 0 = η ξ α t 0. In the following we ssume tht π : E X is vector bundle in the ctegory Mn n with n >. Definition 47. Let α : J X be C 2 -curve. A lift γ Lift E α is α-rllel if α γ = 0. By Eq. 4.0, α-rllelism is loclly equivlent to the first order liner differentil eqution for γ V : γ V = Γ V α V α V,γ V. 4. Definition 48. Given n ffine connection in the tngent bundle of X, C 2 -curve α is sid geodesic if α is α-rllel, tht is α α = 0. Differentil Eqution 4. cn be rewrite for geodesics in the following mnner: α V = Γ V α V α V, α V. 4.2 Theorem 49. Let α : J X be C 2 -curve. Let t 0 J. Given v E αt0, there exists unique lift γ t,v : J E which is α-rllel nd such tht γ t 0,v = v. Proof. It follows from the Theorem of existence nd uniqueness of solutions of liner differentil equtions. See, for exemle, Lng 995, Proosition IV-.9 Definition 50. Let P X := { α : I = [0,] X, α is iecewise C 2} be the sce of ll iecewise C 2 curves of X. A rllel trnsort on vector bundle π : E X is function P : P X Iso E,E q such tht,q X 34

40 4.2 Slitting. P α Iso E α0,e α ; 2. P is rmetriztion indeendent; 3. P α = P α ; 4. P α P β = P βα. A connection in vector bundle E define rllel trnsort on E. In fct, let α : I X be C 2 -curve. By Theorem 49, we cn define ming P α : E α0 E α with P α v := γ,v where γ,v is the unique lift in E which is α-rllel nd γ 0,v = v. P α is liner isomorhism Lng, 995, Theorem VIII-3.4 nd P is rllel trnsort. Theorem 5. Given vector bundle π : E X there exists bijection between covrint derivtives nd rllel trnsorts on E. Proof. See Gibilisco 997, Theorem 6.. Definition 52. A connection is globlly flt if for ny loo σ t ny fixed oint X, P σ = id E. For globlly flt connection, the rllel trnsort P α deends only on the end oints of the curve α, so for ech,q X we cn ut P,q := P α Iso E,E q where α is n rbitrry th from to q. 4.2 Slitting An Orlicz sce s slitting lys n imortnt role in the definition of the connections on the Sttisticl Mnifold M µ so, before introducing them, we shll show how t ech density M µ the tngent sce slits. Definition 53. The closed subsce F of Bnch sce E is sid to be slit if there exists closed subsce G E such tht E = F G. Since continuous liner isomorhism of Bnch sces is homeomorhism, F slits in E iff E is the lgebric direct sum of the closed subsces F nd G. Definition 54. A continuous liner function Π L E; E is rojection if Π 2 = Π. Slittings nd rojections re relted. 35

41 4 Christoffel symbols nd connections Theorem 55. Let E be Bnch sce nd F closed subsce of E. Then F slits iff there exists rojection Π L E; E nd F = {x E : Πx = x} = im Π nd E = F ker Π. Proof. See Abrhm et l. 988, Corollry Proosition 56. Let E be Bnch sce nd λ E. For ech v / ker λ there exists slitting E = ker λ v where v is the -dimensionl liner subsce generted by v. Every w E cn be decomosed s sum w = w λw v + λw v λv λv. 4.3 Proof. Fix v / ker λ nd define the ming i v : R E by R α α v λv v E. It is n injective liner functionl with rnge the closed subsce generted by v. Since i v is continuous, Π := id i v λ L E; E. Liner functionl Π stisfies Π 2 = Π nd it is rojection. Since:. ker Π = v, in fct: w ker Π w = λw v λv v ; 2. im Π = {w E : Πw = w} = ker λ, in fct: w im Π w = Πw = w λw v λv λw = 0; by Theorem 55, E = ker λ v. Definition 57. A Young function Φ : R [0, is n N-function if it is continuous Young function such tht Φ x = 0 iff x = 0, lim x 0 Φ x /x = 0 nd lim x Φ x /x =. Φ,Φ 2,Φ 3 nd x for > re N-functions. Proosition 58. Let X,X, µ be robbility sce, Φ : R [0, be n N- function, then execttion E : L Φ µ R is continuous liner m. Subsce L Φ 0 µ = { u L Φ µ : E u = 0 } is closed. 36

42 4.3 Exonentil Connection Proof. Execttion E is liner. We show it is bounded. Since lim x Φ x /x =, there exists N > 0 such tht for x > N, Φ x > x. Let u L Φ µ \ {0}, u E = u Φ u dµ + u N u u u Φ >N Φ u Φ u N + E Φ N +, u Φ imlies E u C u Φ. L Φ 0 µ is the kernel of E. u u Φ dµ In the cse of Φ = cosh, continuity lso comes from the identity E = DM 0. Proosition 59. Let X,X, µ be robbility sce, Φ : R [0, be n N- function, then the subsce L Φ 0 µ of ll the rndom vribles of L Φ µ hving zero execttion slits in L Φ µ: Projection Π on L Φ 0 µ is L Φ µ = L Φ 0 µ R. Π := id i E where i : R L Φ µ is the inclusion ming. Every f L Φ µ is decomosed s sum f = f E f + E f. Proof. Note tht the set of ll constnt rndom vribles is closed in the Orlicz sce L Φ µ nd it is the rnge of the inclusion ming i : R L Φ µ. Putting v = nd λ = E, everything comes from Proosition 56. Corollry 60. The tngent sce T M µ = B t ech density M µ slits in L Φ µ s L Φ µ = B R. 4.3 Exonentil Connection We shll define n ffine connection in the tngent bundle of the mximl exonentil model E t n rbitrry oint of the exonentil sttisticl mnifold M µ. Definition 6. The exonentil connection in the tngent bundle T E of mximl exonentil model E M µ is the ffine connection such tht, for ech chrt U,s nd for ech u U, the Christoffel symbols Γ e u L 2 B ; B re the null oertor. 37

43 4 Christoffel symbols nd connections The exonentil connection is well defined. Let U,s nd U 2,s 2 be ir of chrts such tht U U 2. We ut f := s 2 e nd F := T s 2 T e = f,l = Df like in the digrm 4.2. For ech u V nd w B L u w = Df u w = w E 2 w. Since DL 0, for ech u V nd w,w 2 B, the chnge of vribles formul 4.3 for the Christoffel symbols simlifies to Γ e 2 f u Df u w,df u w 2 = Df u Γ e u w,w 2 = 0 }{{} =0 nd the Christoffel symbols Γ e nd Γ e 2 glue together in U U 2. Let ξ,η X E be two vector fields nd ξ,η be, resectively, the rincil rt of their locl reresenttion T s s ξ e nd T s η e with resect to chrt U,s : V 4.4 s,ξ e U ξ T U V B T s π U s V By Eq. 4.4 nd since Γ e u 0, the covrint derivtive e ξ : T E T E of the section ξ hs the following locl reresenttion: for ech u,w V B r e ξ u,w = u,dξ u w. Likewise, by Eq. 4.5, the covrint derivtive e ηξ X E hs the following locl reresenttion: V u e ηξ = u,dξ u η u V B. Proosition 62. The exonentil connection ir T E, e is globlly flt nd the ssocited rllel trnsort P e is: P e,q : B w w E q w B q. Proof. Let α : I E be curve of clss C 2 with α 0 = nd α = q. We my ssume tht,q U, the domin of the chrt U,s. For w B, let γ : I T E the lift of α defined by γ t := w E αt w B αt. In the chrt U,s, the rincil rt of the locl reresenttion of γ is the constnt curve γ t = w. Derivtive γ t = 0, so γ stisfies Eq. 4.. The curve γ is α- rllel nd the rllel trnsort P α w = w E q w is indeendent of α. 38

44 4.3 Exonentil Connection The curve α : I U given by α t = e tw Ktw is the geodesic strting in α 0 = with velocity vector α 0 = w B. In fct α t = w E αt w nd α α = 0. e is recisely the exonentil connection introduced by Gibilisco nd Pistone 998, Definition 23 strting from the ssocited rllel trnsort. Since for ech q,q 2 E, L Φ q µ = L Φ q 2 µ we my identify the tngent bundle T E with subset of the Bnch sce L Φ := L Φ q µ for some q E. Exloiting this identifiction, one cn comute exonentil covrint derivtive like in the following roosition Gibilisco nd Pistone, 998, Proosition 25. Proosition 63. Let T E, e be the exonentil connection ir, ξ,η X E, q E nd v B q such tht η q = v, then e η ξ q = d v ξ q E q d v ξ q where d u ξ q := lim h 0 h [ξ σ h ξ σ 0] with σ : ε,ε E C -curve such tht σ 0 = q nd σ 0 = v. Remrk bout Proosition 63. Let ξ X E. We consider ξ : E T E L Φ. Let chrt U,s be given, we ut ξ := ξ e : V L Φ nd σ := s σ. σ ε,ε σ ξ U L Φ e s ξ V If σ 0 = q = e u, σ 0 = v B q nd w = v E v, then we hve Since d v ξ q = lim h 0 ξ σ h ξ σ 0 h Π q : L Φ f f E q f B q is the rojection on B q in the slitting L Φ = B q R, = D ξ u w. e η ξ q = D ξ u w E q D ξ u w = Π q D ξ u w. Tht is, the exonentil covrint derivtive e ηξ cn be viewed s the nturl one which ssigns t ech q E the rojection on B q of the derivtive of ξ : E L Φ t q in the direction η q. 39