Linearization coefficients of Bessel polynomials and properties of Student-t distributions

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1 Lnearzaton coeffcents of Bessel polynomals and propertes of Student-t dstrbutons Chrstan Berg and Chrstophe Vgnat August 5, 5 Abstract We prove postvty results about lnearzaton and connecton coeffcents for Bessel polynomals. The proof s based on a recurson formula and explct formulas for the coeffcents n specal cases. The result mples that the dstrbuton of a convex combnaton of ndependent Studentt random varables wth arbtrary odd degrees of freedom has a densty whch s a convex combnaton of certan Student-t denstes wth odd degrees of freedom. Mathematcs Subject Classfcaton: prmary 33C; secondary 6E5 Keywords: Bessel polynomals, Student-t dstrbuton, lnearzaton coeffcents Introducton In ths paper we consder the Bessel polynomals q n of degree n where q n u α n n n! n The frst examples of these polynomals are α n u, n! n! n! n!!. q u, q u + u, q u + u + u 3. They are normalzed accordng to q n, and thus dffer from the polynomals θ n u n Grosswald s monograph [] by the constant factor n! n! n,.e. θ n u n! n! n q nu.

2 The polynomals θ n are sometmes called the reverse Bessel polynomals and y n u u n θ n /u the ordnary Bessel polynomals. Two-parameter extensons of these polynomals are studed n [], and we refer to ths wor concernng references to the vast lterature and the hstory about Bessel polynomals. For a study of the zeros of the Bessel polynomals we refer to [5]. For ν > we recall that the probablty densty on R f ν x A ν + x ν+, A ν Γν + Γ Γν 3 has the characterstc functon where e xy f ν x dx ν y, y R, 4 ν u ν Γ ν uν K ν u, u, 5 and K ν s the modfed Bessel functon of the thrd nd. If ν n + wth n,,,... then ν u e u q n u, u, 6 and f ν s called a Student-t densty wth ν n + degrees of freedom. For ν then f ν s densty of a Cauchy dstrbuton. Note that for smplcty we have avoded the usual scalng of the Student-t dstrbuton. In ths paper, we provde the solutons of the three followng problems:. Explct values of the connecton coeffcents c n a and ther postvty for a [, ] n the expanson q n au n c n a q u. 7. Explct value of the lnearzaton coeffcents β n a and ther postvty for a [, ] n the expanson q n auq n au n β n aq n+ u Postvty of the lnearzaton coeffcents β n,m a for a [, ] n the expanson q n auq m au n+m n m β n,m aq u. 9

3 Note that β n a β n,n n+ a and that 7 s a specal case of 9 correspondng to m wth c n a βn, a. Note also that u n 9 yelds n+m n m β n,m a, so 9 s a convex combnaton. As polynomal denttes, 7-9 of course hold for all complex a, u, but as we wll see later, the postvty of the coeffcents holds only for a. Because of 4 and 6 formula 9 s equvalent wth the followng dentty between Student-t denstes x a f n+ a a f m+ x a n+m β n,m af + n m x for < a < and s the ordnary convoluton of denstes. Although 9 s more general than 7,8, we stress that we gve explct formulas below for c n a and βn a from whch the postvty s clear. The postvty of β n,m a for the general case can be deduced from the specal cases va a recurson formula, see Lemma 3.4 below. Our use of the words lnearzaton coeffcents s not agreeng completely wth the termnology of [], whch defnes the lnearzaton coeffcents for a polynomal system {q n } as the coeffcents a, n, m such that Snce, n the Bessel case q n uq m u n+m a, n, mq u. q u q u q u + 3q u the lnearzaton coeffcents n the proper sense are not non-negatve. It s nterestng to note that n [3] Koornwnder proved that the Laguerre polynomals L α n u n satsfy a postvty property le 9,.e. n + α u n! n+m L α n aul α n au κ n,m al α u, wth κ n,m a for a [, ] provded α. In ths connecton t s worth pontng out that there s an easy establshed relatonshp between q n and the Laguerre polynomals wth α outsde the range of orthogonalty for the Laguerre polynomals, namely q n u n n L n n u. n 3

4 The problems dscussed have an mportant applcaton n statstcs: the Behrens- Fsher problem conssts n testng the equalty of the means of two normal populatons. Fsher [7] has shown that ths test can be performed usng the d statstcs defned as d f,f,θ t sn θ t cos θ, where t and t are two ndependent Student-t random varables wth respectve degrees of freedom f and f and θ [, π ]. Many dfferent results have been obtaned on the behavour of the d statstcs. Tables of the dstrbuton of d f,f,θ have been provded n 938 by Suhatme [5] at Fsher s suggeston. In 956, Fsher and Healy explcted the dstrbuton of d f,f,θ as a mxture of Student-t dstrbutons Student-t dstrbuton wth a random, dscrete number of degrees of freedom for small, odd values of f and f. Ths wor was extended by Waler and Saw [6] who provded, stll n the case of odd numbers of degrees of freedom, an explct way of computng the coeffcents of the Student-t mxture as solutons of a lnear system; however, they dd not prove the postvty of these coeffcents, clamng only Extensve numercal nvestgaton ndcates also that η for all ; however, an analytc proof has not been found. Ths conjecture s proved n Theorem and 3 below. Secton of ths paper gves the explct solutons to problems, and 3, whereas secton 3 s dedcated to ther proofs. The last secton gves an extenson of Theorem n terms of nverse Gamma dstrbutons. To relate the Behrens-Fsher problem to our dscusson we note that due to symmetry d f,f,θ has the same dstrbuton as t sn θ + t cos θ, whch for θ ], π [ s a scalng of a convex combnaton of ndependent Student-t varables. Usng the fact that the Student-t dstrbuton s a scale mxture of normal dstrbutons by an nverse Gamma dstrbuton our postvty result s equvalent to an analogous postvty result for nverse Gamma dstrbutons. Ths result has been observed for small values of the degrees of freedom n [8]. In [9] the coeffcents are clamed to be non-negatve but the paper does not contan any arguments to prove t. Results. Soluton of problem and a stochastc nterpretaton Theorem. The coeffcents c n a n 7 are expressed as follows c n n a a n n + r n + + r n r a r for n whle c n n a a n. Hence they are postve for a. the collected papers of R.A. Fsher are avalable at the followng address 4

5 A stochastc nterpretaton of Theorem. s obtaned as follows: replacng u by u and multplyng equaton 7 by exp u, we get e a u e a u q n a u n c n a q u e u. Equaton can be expressed that the convex combnaton of an ndependent Cauchy varable C and a Student-t varable X n wth n + degrees of freedom follows a Student-t dstrbuton wth random number K ω + of degrees of freedom: a C + ax n d XKω, where K w [, n] s a dscrete random varable such that Pr{K w } c n a, n.. Soluton of problem and a probablstc nterpretaton Theorem. The coeffcents β n a n 8 are expressed as follows β n n! a 4a a n! n n + n j j j Hence they are postve for a. n n!n +! n!n +! a j. A probablstc nterpretaton of ths result can be formulated as follows. Corollary.3 Let X, Y be ndependent Student-t varables wth n + degrees of freedom, then the dstrbuton of ax + ay follows a Student-t dstrbuton wth a random number of degrees of freedom F ω dstrbuted accordng to.3 Problem 3 Pr {F ω n + + } β n a, n. Theorem.4 The coeffcents β n,m a n 9 are postve for a. We are unable to derve the explct values of the coeffcents β n,m a. However, ther postvty allows us to clam the followng Corollary. Corollary.5 Let X, Y be ndependent Student-t varables wth resp. n +, m + degrees of freedom, then the dstrbuton of ax + ay follows a Student-t dstrbuton wth a random number of degrees of freedom F ω dstrbuted accordng to Pr {F ω + } β n,m a, n m n + m. 5

6 Theorem.6 For let n,..., n be nonnegatve ntegers and let a,..., a be postve real numbers wth sum. Then q n a uq n a u q n a u L β j q j u, u R, jl where the coeffcents β j are nonnegatve wth sum, l mnn,..., n and L n + + n. 3 Proofs 3. Generaltes about Bessel polynomals As a preparaton to the proofs we gve some recurson formulas for q n. They follow from correspondng formulas for θ n from [], but they can also be proved drectly from the defntons and. The formulas are We can wrte q n+ u q n u + q nu q n u u n n u 4n q n u, n, 3 u n q n u, n. 4 δ n q u, n,,... 5 and δ n s gven by a formula due to Carltz [6], see [, p. 73] or [6]: δ n { n+! n n! n!!+ n! for n n for < n. 6 Later we need the followng extenson of 3 whch we formulate usng the Pochhammer symbol z n : zz + z + n for z C, n,,.... Lemma 3. For n we have u q n u γ n, q n+ u where γ n, n + + n. 7 Proof: The Lemma s trval for and reduces to the recurson 3 for wrtten as u q n u n q n+ u q n u. 8 6

7 We wll prove the formula 7 by nducton n n, so assume t holds for some n and all n. Multplyng the formula of the lemma by u we get hence by 8 u + q n u γ n, u q n+ u, u + q n+ + u γ n, n + + [q n++ u q n++ u] γ n, n + + q n++ u [ ] + n + + γ n, n + γn, n q n++ u γ n, n + q n+ u. Usng the nducton hypothess we easly get and γ n, n n + + γ n+,+ +, γ n, n + 3 n + + n + + n 3 + γ n+,+. Concernng the coeffcent C to q n++ u above we have [ C + n + + n + + n +n + ] n + + n n n + ++ n [ ] n n [ ] + + n + ++ n n n + ++ n 3 + γ n+,+. We stress that Lemma 3. s the specal case ν n + of the followng recurson for modfed Bessel functons of the thrd nd. Lemma 3. For all ν > and all nonnegatve ntegers j < ν we have for u > j j u ν+j K ν j u j Γν + Γν + j uν+ K ν+ u 7

8 and u j ν j u j j j j Γ ν + Γ ν + Γ ν + j Γ ν j ν+ u. Proof: The second formula follows from the frst usng formula 5, and the frst can be proved by nducton usng the followng recurson formula for modfed Bessel functons of the thrd nd, cf. [7, p. 79] We sp the detals. 3. Proof of Theorem. From and 5 we get wth c n a where q n au n j K ν u K ν+ u ν u K νu. n j a j α n j δ j a n!! n!! α n j a j j n j,j + δ j q u n In partcular c n n a a n and for n We clearly have wth pa p r c n a a n! n! n + pa n + r c n aq u a j n j! j + n j! j! + j!! pa, 9! n! + + a n!! +!. n + r r pr r! a r n! + + n! r! +! and we only consder r n +. To sum ths we shft the summaton by r. For smplcty we defne T : n r + r and get r p r T n r! + r + + n r!! + r!. 8

9 We wrte + r r and splt the above sum accordngly r p r + T n r! n r!! + r! T n r! n r!! r!. Note that for nonnegatve ntegers a, b, c wth b, c a we have b c a! b!c!! a! b!c! b c b c a! a! b!c! F b, c; a;, where we use that the sum s an F evaluated at. Its value s gven by the Chu-Vandermonde formula, see [], hence b c a! b!c!! a!c a b a b b!c!. The two sums above are of ths form and we get where r p r n r! n r! + r! Q, Q + n + n r + r n n r + r n n r + r n n r n + n r + r n n r [ + r n + r n] rn + n + r + n r n + n r, where we used a n n a n n twce. Ths gves and fnally n + n! r p r r + r n r! pa n + r a r r r! n + n!. + r n r! Note that the term correspondng to r s zero. If we nsert ths expresson for pa n 9, we get the formula of Theorem.. Remar 3.3 The evaluaton above of r p r can be done usng generatng functons le n [6]. The authors want to than Mogens Esrom Larsen for the dea to use the Chu-Vandermonde dentty twce. 9

10 3.3 Proof of Theorem. The startng pont s the followng formula of Macdonald, see [7] K ν zk ν X exp[ s z + X s ]K ν zx s ds s, whch we wll use for ν n +, z au, X au. Multplyng by ν a au ν Γν and usng 5 we fnd ν au ν au ν exp [ s Γν u a + a ] a au s ν ν ds. s s We now nsert that wth ν n + we have ν u e u q n u and hence after some smplfcaton n+ Γn + e u q n a u q n a u ] s u exp [ s n a au qn ds. s s We next nsert the expresson for q n under the ntegral sgn. Ths gves n e u q n a u q n a u α n a a u n+ Γn + Usng the followng formula from [, 3.479] x ν exp βx γx dx ] s u exp [ s n s β ν/ K ν βγ γ ds. and agan 5 the above s equal to n α n n + Γn + a a n+ Γn + e u u q n u. Fnally, usng Pochhammer symbols and sppng absolute values snce we are now dealng wth a polynomal dentty, we get q n auq n au n α n a a n u q n u. n

11 α n Usng the expresson for u q n u from Lemma 3. and the expresson for n we then get q n auq n au n n n n n a a n! n n q n+ u a a n γ n, q n+ u n n! γn, hence q n auq n au n β n aq n+ u wth n n β n a a a n n n! n + + n n n a a a a l +l n l + l n +l n + l! +l Collectng we get β n l n l + l+ n l n ln l + l+ n+ n n a a a a a l l +l n +l n+ n +l n! l! l a a n! n+ n n! 4a a l n l! n l n! n l! l! n! a a n +! n! n +!! l n l a l n l! n l. n l! l! Expandng a l usng the bnomal formula and nterchangng the sums we get n a l n l! n l n l!l! l n j a j j! j n lj n l! n l. n l!l j!

12 We clam that S : n n l! n l n l!l j! lj n! n + n j n j j!! n! j. 3 and we have obtaned the fnal formula β n n! a 4a a n! n n!n +! n!n +! n n + n j j j a j. We wll see that 3 s a Chu-Vandermonde formula. summaton ndex puttng l j + m we get In fact, shftng the S n j n j m n j m! n + j m, m!n j m! and callng the general term n ths sum c m we get c m+ m + + j nm + j n, c m m + m + + j n whch shows that the sum s a F. We have S n j c F n j, j n ; + j n;, and usng the Chu-Vandermonde dentty, cf. []: we get S n j n j F n, a; c; c a n c n, n j! n + + n j n j! + j n n j n j! n j! j + n j, n + n j where we used a n n a n n twce. We can now smplfy to get S n n! n j n j!, j and applyng the formula p p! p! p

13 twce we get S Now we can wrte n! n! n n n! j! j j n j! j!. n j j j n +!n j! n!n j +!, and hence S n! n + n j j!! n! j n j. 3.4 Proof of Theorem.4 For n, m and a R, we can wrte q n au q m a u m+n β n,m a q u 4 for some unquely determned coeffcents snce the left-hand sde s a polynomal n u of degree n + m. Clearly β n,m a s a polynomal n a satsfyng β n,m a β m,n a. 5 We shall prove that β n,m a for a and that β n,m a f < n m, whch wll be a consequence of the followng recurson formula. Lemma 3.4 For n, m, we have + βn,m + a a n βn,m a a + m βn,m a, 6 where,,..., m + n. Furthermore β n,m a. Proof: Dfferentatng 4 wth respect to u gves aq n au q m a u + a q n au q m a u m+n and usng the formula 4 we fnd a q n au au n q n au q m a u + a q n au q m a u au m q m a u m+n βn,m a q u u q u 3 β n,m a q u

14 and usng 4 once more we get a u n q n au q m a u a u m q n au q m a u β n,m a u n+m β n,m + a + q u. For u ths gves β n,m a and dvdng by u and equatng the coeffcents of q u, we get the desred formula. Now the proof of Theorem.4 s easy by nducton n and by the symmetry formula 5, we can assume n m. Let a. We prove that β n,m a for n + m and that t s zero for < m under the assumpton n m. Ths s true for by Lemma 3.4 when m, and for m t follows by Theorem.. Assume now that the results hold for and assume + n + m. The nonnegatvty for + now follows by Lemma 3.4, and lewse f + < m n the coeffcent s snce < n m. 3.5 Proof of Theorem.6 By Theorem.4 the result holds for. Assumng t holds for we have q n a u q n a u L jl γ j q j a u, u R 7 wth l mnn,..., n, L n + + n and γ j because we can wrte a j u a j a u, j,...,. a If we multply 8 wth q n a u we get L jl γ j q n a uq j a u and the asserton follows. L n +j γ j jl n j 4 Inverse Gamma dstrbuton β n,j a q u, Grosswald proved [] that the Student-t dstrbuton s nfntely dvsble. Ths s a consequence of the nfnte dvsblty of the nverse Gamma dstrbuton because of subordnaton. It was proved later that the nverse Gamma dstrbuton s a generalzed Gamma convoluton n the sense of Thorn, whch s stronger than self-decomposablty and n partcular stronger than nfnte dvsblty, see e.g. Bondesson [4] and the recent boo by Steutel and van Harn [4]. The followng densty on the half-lne s an nverse Gamma densty wth scale parameter 4 and shape parameter ν > : 4

15 C ν exp 4t t ν, t >, C ν ν Γν. 8 Let the correspondng probablty measure be denoted γ ν and let further g t x 4πt exp x 4t, t >, x R denote the Gaussan semgroup of normal denstes n the normalzaton of [3]. Then the mxture f ν x g t x d γ ν t 9 s the Student-t densty 3 wth ν degrees of freedom. The correspondng probablty measure s denoted by σ ν. Ths formula says that σ ν s subordnated to the Gaussan semgroup by an nverse Gamma dstrbuton, and t mples the nfnte dvsblty of Student-t from the nfnte dvsblty of nverse Gamma. Snce the Laplace transformaton s one-to-one, t s clear that f two probabltes γ, γ on ], [ lead to the same subordnated densty g t x dγ t g t x dγ t, x R, then γ γ. If we denote τ a x ax, the dstrbuton τ a σ τ a σ s gven n n+ m+. However note that τ a g t xdx g ta xdx, so hence τ a σ ν τ a σ ν τ a σ ν g ta xd γ ν t dx, 3 g ta dx g s a dx d γ ν td γ ν s g ta +s a dx d γ ν td γ ν s g u x dτ a γ ν τ a γ ν u dx. Therefore, usng 3 we see that for ν n+, ν m+ wth n, m,,... the formula rewrtten as s equvalent to τ a σ τ a σ n+ m+ τ a γ τ n+ a γ m+ 5 n+m n m n+m n m β n,m aσ + β n,m a γ. 3 +

16 Ths shows that Theorem.4 s equvalent to the followng result about nverse Gamma dstrbutons: The dstrbuton of a Z n + a Z m, where Z n, Z m are ndependent nverse Gamma random varables wth dstrbuton 8 for ν n+, m+ respectvely, has a densty whch s a convex combnaton of nverse Gamma denstes. Ths result can be extended to the multvarate Student-t dstrbutons as follows. A rotaton nvarant N varate Student-t probablty densty s gven for x x,..., x N R N by where f N,ν x A N,ν + x ν N, A N,ν Γ ν + N Γ ν Γ, N x, y N x y, x x, x, x, y R N. It s easy to verfy that f N,ν x s subordnated to the N-varate Gaussan semgroup g N,t x 4πt N exp x, t >, x R N 4t by the nverse Gamma densty 8,.e. f N,ν x Therefore the characterstc functon s gven by generalzng 4. In fact R N e x,y f N,ν x dx g N,t x d γ ν t. 3 R N e x,y f N,ν x dx ν y 33 e x,y g N,t x dx d γ ν t R N e t y d γ ν t and the result follows by. As a concluson, the Theorems.,. and.4 apply n the multvarate case. For example, an equvalent form of wrtes as follows: wth < a <, a N f N,n+ a x a N f N,m+ a x n+m n m β n,m a f N,+ x. 6

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18 [6] G.A. Waler, J.G. Saw, The Dstrbuton of Lnear Combnatons of $t$- Varables. Journal of the Amercan Statstcal Assocaton, Vol. 73, Issue 364, Dec. 978, [7] G.N. Watson, A Treatse on the Theory of Bessel Functons, nd edton. Cambrdge Unversty Press, Cambrdge 966. [8] V. Wtovsý, Exact dstrbuton of postve lnear combnatons of nverted ch-square random varables wth odd degrees of freedom. Statstcs & Probablty Letters 56, Department of Mathematcs, Unversty of Copenhagen, Unverstetsparen 5, DK-, Copenhagen, Denmar. Emal: berg@math.u.d Laboratore d Informatque IGM, UMR 849, Unversté de Marne-la-Vallée, 5 Bd. Descartes, F Marne-la-Vallée Cedex, France. Emal: vgnat@unv-mlv.fr 8