GENERALIZED CAUCHY-STIELTJES TRANSFORMS OF SOME BETA DISTRIBUTIONS

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1 Communications on Stochastic Analysis Vol. 3, No. 009) 97-0 Serials Publications GENEALIZED CAUCHY-STIELTJES TANSFOMS OF SOME BETA DISTIBUTIONS NIZA DEMNI Abstract. We express generalized Cauchy-Stieltjes transforms GCST) of some particular beta distributions depending on a positive parameter as -powered Cauchy-Stieltjes transforms CST) of some probability measures. The CST of the latter measures are shown to be the geometric mean of the CST of the Wigner law together with another one. Moreover, these measures are shown to be absolutely continuous and we derive their densities by proving that they are the so-called Markov transforms of compactly-supported probability distributions. Finally, a detailed analysis is performed on one of the symmetric Markov transforms which interpolates between the Wigner = ) and the arcsine = ) distributions. We first write down its moments through a terminating series 3 F and show that they are polynomials in the variable /, however they are no longer positive integer-valued as the are for =, for instance = ). Second, we compute the free cumulants in the case when = and explain how to proceed in the cases when = 3, 4. Problems of finding a deformation of the representation theory of the infinite symmetric group and an interpolating convolution are discussed.. Motivation Let > 0 and µ a probability measure possibly depending on ) with finite all order moments. The generalized Cauchy-Stieltjes transform GCST) of µ is defined by z x) µ dx) for nonreal complex z lying in a suitable branch [8, 7, 0]. For =, it reduces to the ordinary) Cauchy-Stieltjes transform CST) which has been of great importance during the two last decades for both probabilists and algebraists due the central role it plays in free probability and representation theories [, 0]. Moreover, CST were extensively studied and they are handable in the sense that for instance, a complete characterization of those functions is known and one has a relatively easy inversion formula due to Stieltjes [0]. However, their generalized versions are more hard to handle as one may realize from the complicated inversion formulas displayed in [8, 7, 0]. In this paper, we adress the problem of relating 000 Mathematics Subject Classification. Primary 44A5; 44A35; Secondary 44A60. Key words and phrases. Generalized and ordinary Cauchy-Stieltjes transforms, Markov transforms, beta distributions, Wigner and arcsine distributions. *This research was supported by SFB

2 98 NIZA DEMNI both generalized and ordinary transforms, that is, given µ check whether there exists a probability measure ν such that [ ] z x) µ dx) = z x ν dx).) for z in some suitable complex region we shall consider the principal determination) and characterize ν in the affirmative case. Doing so will lead for instance to the invertibility of [ ] / z z x) µ dx) for z belonging to some neighborhood of infinity [] and as a by-product to a kind of -free cumulants generating function for µ, referring to the case = [9]. However, this may not be always possible as we shall see, that is, ν may not be a probability measure for some. Moreover, when ν is shown to be a probability measure, it is not easy to check whether it is absolutely continuous or not and more harder will be to write down its density when it is so. This was behind our willing to get an insight into the above problem through GCST a particular class of probability measures µ [7]. The latter probability measures may be mapped via affine transformations into beta distributions β a,b dx) = x) a + x) b [,] x)dx with parameters a, b > depending on, and their monic orthogonal polynomials, say P n ) n, are the only ones up to a conjecture, see [7]) that admit ultraspherical-type generating functions: n 0 ) n Pn x)z n = n! u z)f z) x), x suppµ ),.) where u, f satisfy some technical conditions in a suitable open complex region near z = 0 so that.) makes sense. Due to the orthogonality of Pn, n 0, one gets after integrating both sides in.) with respect to µ this is a special case of the multiplicative renormalization, [5]): u z) = f z) x) µ dx), for small z. Thus, if f is invertible, the last equality is equivalent to u [f z)] = z x) µ dx) so that u f ) is the GCST of µ. Fortunately, as one easily sees from the reminder below, f may be identified with the CST in some neighborhood of infinity) of a semi-circle law of mean and variance depending on.

3 GCST FO BETA DISTIBUTIONS 99. eminder and esults Assume µ has zero mean and unit variance, then µ has a generating function for orthogonal polynomials of ultraspherical-type if and only if it is compactlysupported and belongs to one of the four families corresponding to [7]: u z) = z, f z) = + z + z, > 0, z u z) = /)z, f z) = z + z, > / z u z) = ± z/, f z) = z ± + z, > /. We did not displayed the density of µ for sake of clarity and we will do it later. Nevertheless, it is worth recalling that the first family corresponds to the monic Gegenbauer polynomials Cn) n, the second one corresponds to Cn ) n and is related to the Poisson kernel, while the remaining families correspond to shifted Jacobi polynomials whose paremeters differ by. Now, let G m,σ be the CST of a semi-circular law of mean m and variance σ [4], then its inverse in composition s sense, say K m,σ, is given by [4] K m,σ z) := G m,σz) = σ z + m + z..) Hence, the reader can check that f fits K m,σ for particular means and variances depending on, so that f fits G m,. Nevertheless, we will only make use of G := G 0, since the elemantary identity holds G m,σ z) = ) x m σ G. σ Accordingly, our first main result may be stated for sufficiently large z as [ ] / z x) β a,bdx) = [ũ Gz))] / = G α) z) Gz) γ),.) where α) + γ) = and ũ z) = z, α) =, ũ z) = z z, α) =, Gz) = z 4, ũ z) = z ± z, α) =, Gz) = z ± for the four families respectively. It follows from the Nevannlina s characterization of CST of probability measures that [ ] / z z x) β a,bdx) defines a CST of some probability measure ν for provided that 0 α) = γ). Note that under this condition, the CST of ν is the geometric mean of G, G and that one discards the values ]/, [ for the second family The third and fourth families correspond to the plus and minus signs respectively.

4 00 NIZA DEMNI for which G is the CST of the arcsine distribution [9]. The second main result shows that, if 0 α) = γ), then ν are absolutely continuous probability measures and gives their density. This follows from the fact that ν is the so-called Markov transform of some compactly-supported probability measure τ [0], that is z x ν dx) = exp logz x)τ dx),.3) and from Cifarelli and egazzini s results see [0] p. 5). For the first and the second families, ν is given by the Wigner distribution and a symmetric deformation of it respectively, while for the remaining ones, it is a nonsymmetric deformation of the Wigner distribution. Since the latter is a universal limiting object representation theory of the infinite symmetric group, spectral theory of large random matrices, free probability theory), we give a particular interest in ν corresponding to the second family. We first express its moments by means of a terminating 3F series interpolating between the moments of the Wigner and the arcsine distributions Catalan and shifted Catalan numbers, [0] p.64). Unfortunately, the moments are no longer positive integer-valued as it is shown for =. Nevertheless, they are polynomials in the variable /. Finally, we use.) to compute the inverse of its CST and the free cumulants generating function in the case = which involve a weighted sum of the Catalan and the shifted Catalan numbers. For = 3, 4, this is a more complicated task and by Galois theory, this is a limitation rather than a restriction since we are led to find a root of a polynomial equation of degree. emarks.. ) For the second family µ and for the discarded values of such that the condition 0 α) = γ) fails to hold, [ z ] / z x) µ dx) does not define the CST of a probability distribution. ) It follows from.) and.3) that [ ] z x) µ dx) = exp logz x)τ dx) = z x ν dx). Similar identities already showed up in relation to Bayesian statistics see [0] p. 59) and in relation to the so-called GGC random variables and Dirichlet means [9]. Throughout the paper, computations are performed up to constants depending on, which normalize the finite positive measures involved here to be probability measures. The paper is divided into five sections: the first four sections are devoted to the four families µ while the last one is devoted to the particular interest we give in the probability measure ν corresponding to the second family.

5 GCST FO BETA DISTIBUTIONS 0 3. Markov Transforms: Symmetric Measures 3.. First family. On the one hand, µ dx) x + ) ) / [± x)dx, > 0, +)] and its image of under the map x + )/x has the density proportional to ) / [,] x). On the other hand, it is easy to see that so that It follows that f z) = + z + z = K z) 0, +)/ ) f z) = G z) = 0, +)/ + G + z. z x) ) / dx = [Gz)] = [ dx ]. z x π Using the fact that the Wigner distribution is the Markov transform of the arcsine distribution see [0] p.64), one finally gets Proposition 3.. For > 0, z x) µ dx) = exp where [ logz x)τ dx) = µ dx) ) / [,] dx, ν dx) = [,] dx, π τ dx) = π [,]dx. ] z x ν dx). 3.) emark 3.. The value = / corresponds to the uniform measure on [, ]. ecently, it was shown that this measure is the only nonatomic probability measure with a square root-type generating funtion [6]. 3.. Second family. The density of µ reads ) 3/ µ dx) x [, ] x)dx, > /, and we map it using x /x to ) 3/ [,] x)dx, > /. Now, ) f z) = G z

6 0 NIZA DEMNI so that Using Gz) = ) 3/ G z) dx z x) G z) = ũ G)z). dx z x π = z z 4 = z + z 4, for z C \ [, ] together with G z) + = zgz), then G z z) = zgz) = z + z 4 = z 4Gz). But dx π z x = z 4 = exp log[z )z + )] for suitable z, therefore Proposition 3.3. ) 3/ G z) dx z x) z 4 = G z)g arcsin z) = exp logz x)τ dx) where τ dx) = ) π [,]dx) + δ + δ dx), which is not a probability measure unless. Next, let us seek ν, such that exp logz x)τ dx) = z x ν dx). In fact, T z) := G / z)g / arcsin z), z C \ [, ], defines the CST of a probability measure and this claim is readily checked using Lemma II.. in [8]. More precisely, one has for Iz) > 0 arg[g / z)g / arcsin z)] = ) arg[gz)] + arg[g arcsinz)] ] π, 0[ so that T is of imaginary-type maps the upper half-plane into the lower halfplane), and lim iyt iy) = lim y y [iygiy)] / [iyg arcsin iy)] /. For / < <, one easily gets the inequality arg[g arcsin z)] < arg[t z)] < arg[g arcsin z)] arg[gz)] and there is no guarantee for ν to be a probability distribution. In order to characterize ν,, the Markov transform of τ, we will use results by Cifarelly

7 GCST FO BETA DISTIBUTIONS 03 and egazzini [0]. In fact, since τ is a compactly-supported probability measure, then ν is absolutely continuous with density proportional to where sinπf x)) exp log x u τ du), 3.) πf x) := τ ], x]) 0, if x <, π/), if x =, = /)[arcsinx/) + π/], if x [, [, π, if x. Note that since 3.) is valid when τ is the arcsine distribution and ν is the Wigner distribution, one deduces that exp log x u π du, x [, ], 4 u does not depend on x is constant). This striking result may be used to derive the density of ν in our case and in the forthcoming ones since τ is a convex linear combination of the arcsine distribution and a discrete probability measure. In the case in hand, easy computations yield Proposition 3.4. ν dx) dx cos [ ) arcsin x ] ) ],[x),. / Note that = corresponds to the arcsine distribution while = correponds to the Wigner distribution. Thus, ν interpolates between them. The reader may wonder how to compute the normalizing constant or the moments of ν. This will be clear after dealing with the two remaining families µ. 4. Markov Transforms: Nonsymmetric Measures 4.. Third family. The probability distribution µ has the density where x x ) / + ) 3/ x, [ ] +,, > /. Moreover f z) = z + + z = K /,/ z).

8 04 NIZA DEMNI Thus, [ z) = G /,/ z) = G [ z = G f z )] Now, the image of µ under the map x x )/ transforms its density to x ) / + x ) 3/ [,] x) and one easily sees that z x) ) 3/ x) dx G z) Gz) := ũ G)z). Now note that Gz)) = + G z) Gz) = z )Gz) which yields Gz) = z Gz) where the branch of the square root is taken so that G is of positive imaginary-type i.e., maps the upper half plane into itself). Proposition 4.. ) 3/ G z) x) dx z x) Gz) = G / z). z In this case τ satisfies for suitable z logz x)τ dx) = whence we deduce that τ dx) = ]. ) loggz)) log z ) π [,]dx) + δ dx) which is a probability measure for all > /. Besides, the same arguments used before show that for suitable z z G /) z) z ) /, is the CST of a probability distribution ν, > / which is absolutely continuous with density given by Proposition 4.. ν dx) dx sin [ ) arcsin x + π ) ] x) ],[x), > /. /

9 GCST FO BETA DISTIBUTIONS Fourth family. The density of µ is given by x + ) 3/ + ) / x + 4.) where x [ ] +,, > /. The density of the image of µ under the map x x + )/ reads x ) 3/ + x ) / [,] x) ) 3/ + x) [,] x). The same scheme used to deal with the third family gives τ dx) = ) π [,]dx) + δ dx) 4.) and that the CST of ν is given by G /) z) z + ) /. However, the density of ν is somewhat different from the one in the previous case: [ ν dx) cos ) arcsin x ] dx + x) ],[x), > /. 4.3) / Proposition 4.3. Let > /. Then [ ] z x) µ dx) = exp logz x)τ dx) = z x ν dx) where µ, τ, ν are displayed in 4.), 4.) and 4.3). emarks 4.4. ) For the four families, the measure τ may be mapped to π x [,]dx) + M δ dx) + N δ dx) for some positive constants M, N. A more wider class of measures including the above one were considered in [3]. ) For 0 < a <, b = a, one can always define an operation ν, ν ) ν/g ν = G a ν G b ν, where ν, ν, ν are probability measures. However, every probability measure will be idompotent and the operation is not commutative unless a = b = /.

10 06 NIZA DEMNI 5. On the Moments of the Second Markov Transform It is known that the Wigner distribution is a universal limiting distribution: it is the spectral distribution of large rescaled random matrices from the so called Wigner ensemble, the limiting distribution of the rescaled Plancherel transition of the growth process for Young diagrams [] and more generally of the rescaled transition measure of rectangular diagrams associated with roots of some adjacent orthogonal polynomials []. It also plays a crucial role in free probability theory where it appears as the central limiting distribution of the sum of free random variables [, 9]. Note also that the standard arcsine distribution is the central limiting distribution for the t-convolution with t = / [3, 5, 4] and is the limiting distribution of the so-called Shrinkage process []. As a matter a fact, it is interesting to find a parallel to the above facts when the Wigner law is replaced by the Markov transform ν corresponding to the second family. More interesting is to define a convolution operation that interpolates the t-convolutions for t = / = ) and t = =, free convolution) having ν as a central limiting distribution [6]. Since ν is symmetric and compactly-supported, it is entirely determined by its even moments and we claim Proposition 5.. The normalizing constant of ν is given by c = / Γ /))Γ3/ /)) π Γ /) and the even moments of ν may be expressed as ) m n := x n ν dx) = n n, /), 3/ /) 3F ;. /, Moreover, the moments are polynomials in the variable /. Proof. : make the change de variable x sin x in the integral [ x n cos ) arcsin x ] [ ] /) dx, n 0, to obtain Then expand n+ / π/ 0 [ [sin x] n cos ) ] x [cos x] / dx, n 0. [sin x] n = cos x) n = and use the formula see [0], p. 77) π/ 0 n k=0 ) n ) k [cos x] k k cos[p q)x][cos x] p+q π Γp + q ) dx = p+q, p + q >, 5.) Γp)Γq)

11 GCST FO BETA DISTIBUTIONS 07 with p + q = k + 3 /, p q = /, to get x n [ cos ) arcsin x ] dx [ ] /) n ) n Γk + /) = n π k. k Γk + /)k! 4) This may be expressed through 3 F hypergeometric series as follows: write the binomial coefficient as n ) n! = k k!n k)! = n) )k k k! and use the duplication formula to rewrite πγ k + ) = k+ / Γ It follows that x n cos [ ] / k=0 k + [ ) arcsin x ] dx To prove the last claim, let y = / and expand: y ) = k y ) k y ) k = and similarly 3 y ) k ) Γ k + 3/ ). ) = c n n, /), 3/ /) 3F ;. /, y ) k y )y 4) y k + )y k), = ) k y 3)y 5) y k ), y) k = ) k y )y 3) y k ). ) The proof ends after forming the ratio y/) k 3/ y/) k = k y k ) y k)y k ) y) k 4) so that m n := p n y) = n ) n n k y k ) y k)y k ) 4. k k! k=0

12 08 NIZA DEMNI emark 5.. The exponential generating series of p n ) n is given by p n y) zn n! = 4 n k y k ) y k)y k ) z n n k)!k! k! n 0 0 k n = e 4z y k ) y k)y k ) z k k! k! k 0 ) = y e 4z y/, 3 y)/, F ; 4z. y, 5.. Combinatorics. For =, one recovers the moments of the arcsine distribution given by use duplication formula) ) n = n /) ) n = n n, / F ;. n ) n When =, one recovers the moments of the Wigner distribution: ) n = n /) ) n = n n, 3/ F ;. n + n ) n Both moments are integers and it is known that they count the shifted and the ordinary Catalan paths respectively see [0], p. 64). Unfortunately, m n is not positive integer-valued in general, nevertheless one gets for = m n = n+ n k=0 ) k n k ) π Γk + 3/) Γk + ) n n = n ) k k k=0 ) ) 4k + k + 4k so that the few first even moments are equal to, 3/, 3/8, 87/6, 4859/ Inverses of CST for =, 3, 4. Let = n be a positive integer. ecall that [ ] n [G n z)] n := z x) ν ndx) = ũ n Gz)), where as before G is the CST of the Wigner distribution. For the first family, G n = G for all n so that K n = K and Kz) = z + /z for z in a neighborhood of zero. For the remaining families, the inverse of G n is given by G n z) := K n z) = Kũ n z n )), K := G, for small z, subject to the condition K n z) = z + nz) = z + k 0 r k z k where n is an entire function known in free probability theory as the free cumulants generating function of ν n, r k ) k is the sequence of the free cumulants, r 0, r are the mean and the variance of ν n respectively. For the second family, one has to invert z ũ n z) = zn z for small z in the lower half unit disc image of G). We are thus led to find one root of the polynomial z n +z w w for complex numbers z, w in a neighborhood of zero. This task is very complicated since Galois theory asserts that the roots cannot be

13 GCST FO BETA DISTIBUTIONS 09 expressed by means of radicals unless n 4. For n =, easy computations show that K z) = z + z + z = z + /) k k + )! )k k + 3/)z k+. k 0 Thus one sees that r 0 = 0, r = 3/ which agrees with our above computations, and that [ r k+ = ) k /)k + ] /) k. ) k ) k Thus, ) k k+ r k+ = k+ /) k ) k + k /) k ) k = k k ) + k + ) k. k For n = 4, one has a quadratic polynomial and setting v = z, one is led to v + wv w = 0 so that v = w + w + 4w. The case z = 3 is more complicated and we supply one way to adress it: make the substitution z = Z w/3 for suitable Z to get z 3 + wz w = Z 3 w 3 Z + 7 w3 w. The last polynomial has the same form as a + b) 3 3aba + b) a 3 + b 3 ) = 0 which hints to look for a root of the form Z = a + b where [ ] ab = w 3, a3 + b 3 = w 7 w, and computations are left to the curious reader. emark 5.3. The above line of thinking remains valid for the nonsymmetric Markov transforms for which ũ n z) = zn ± z. For = 3, one already has the appropriate form of the polynomial and there is no need to make the above substitution. However, the case = 4 needs more developed techniques since the polynomial is of degree four and is not quadratic. Acknowledgment. This work was suggested by Professor M. Bozejko of the Institute of Mathematics, Wroclaw University to whom the author is grateful. The author finished the paper while visiting Georgia Institute of Technology and University of Virginia. He wants to thank Professors H. Matzinger and C. Houdre, and the administrative staff for their hospitality. A special thank is given to Professor C. F. Dunkl for fruitful discussions held at University of Virginia.

14 0 NIZA DEMNI eferences. Biane, P.: epresentations of symmetric groups and free probability. Adv. Math. 38. no.. 998), Bercovici, H. and Voiculescu, D. V.: Lévy-Hincin-type theorems for multiplicative and additive free convolutions. Pacific. J. Math ), Bozejko, M.: Deformed Free Probability of Voiculescu,. I. M. S. Kokyuroku ), Bozejko, M. and Demni, N.: Generating functions of Cauchy-Stieltjes type for orthogonal polynomials. Infinite. Dimen. Anal. Quantum Probab. elat. Top., no., 009), Bozejko, M., Krystek, A. D., and Wojakowski, L. J.: emarks on the r and convolutions, M. Zeit. 53. no.. 006), Cabanal-Duvillard, T.: Un théorème central limite pour des variables aléatoires noncommutatives. C.. A. S. t. 35, Série I. 997), Demni, N.: Ultraspherical type generating functions for orthogonal polynomials. To appear Probab. Math. Statist. 9. no.. 009) 8. Hirschmann, I. I. and Widder, D. V.: Generalized inversion formulas for convolution transforms. II. Duke Math. J ), James, L. F., oynette, B., and Yor, M.: Generalized Gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probability Surveys. 5, 008), Kerov, S.: Interlacing measures. Amer. Math. Soc. Transl. Ser ), Kerov, S.: Transition probabilities of continuous Young diagrams and Markov s moment problems. Func. Anal. Appl. 7. no.. 993), Kerov, S.: Asymptotic separation of roots of orthogonal polynomials. Algebra and Analysis. 5. no ), Koornwinder, T.: Orthogonal polynomials with weight function x) α + x) β + Mδx + ) + Nδx ). Canad. Math. Bull. 7. no.. 984), Krystek, A. D. and Yoshida, H.: The combinatorics of the r-free convolution. Infinite Dimen. Anal. Quantum. Probab. 6. no ), Kubo, I., Kuo, H.-H., and Namli, S.: The characterization of a class of probability measures by multiplicative renormalization. Commun. Stoch. Anal., no ), Kubo, I., Kuo, H.-H., and Namli, S.: Applicability of multiplicative renormalization method for a certain function. Commun. Stoch. Anal., no ), Schwarz, J. H.: The generalized Stieltjes transform and its inverse. J. Math. Phys. 46. no.. 005), 0350, 8pp. 8. Shohat, J. A. and Tamarkin, J. D.: The Problem of Moments. American Mathematical Society, New York, Speicher,.: Combinatorics of Free Probability Theory. Lectures Institut Henri Poincaré, Paris, Sumner, D. B.: An inversion formula for the generalized Stieltjes transform. Bull. Amer. Math. Soc ), Nizar Demni: Fakultät für mathematik, SFB 70, universität Bielefeld, Bielefeld, Germany. address: demni@math.uni-bielefeld.de.