(), 1–27 /.. A probabilistic counterpart of Askey scheme for continuous polynomials Jean-Renaud Pycke Abstract. Random variables corresponding to weight functions of the Askey scheme for continuous polynomials are introduced. Beside normal, Beta and Gamma distributions four new families of real random variables corresponding to Wilson, continuous dual Hahn, continuous Hahn and Meixner-Pollaczek polynomials are discussed. Formulas for the moments and asymptotic behavior of the tails of the distributions are provided. Keywords. Askey scheme, orthogonal polynomials, parametric model. 2010 Mathematics Subject Classification. 42C05, 60E05. 1 Introduction The main goal of this paper is to introduce families of real random variables de- rived in a natural way from the orthogonal families of the Askey-scheme for con- tinuous polynomials. Both graphs (1.9) and (1.10) can be seen as probabilistic counterparts of the Askey scheme for continuous polynomials (1.1). The Askey scheme for continuous and discrete polynomials represents an out- standing achievement in the theory of special functions over the last decades. The Askey-scheme for continuous polynomials encompasses the theory of classical orthogonal polynomials of Laguerre, Hermite and Jacobi. It provides relations between these celebrated sequences of polynomials and includes them in a much wider class. The latter is much less familiar to non specialists and contains Wat- son, continuous dual Hahn, continuous Hahn, Meixner-Pollaczek polynomials. Whereas the connection between classical polynomials and parametric families of random variables as Gamma, Beta or Gaussian distributions are well-known, the random variables corresponding to this wider family have not been studied in a systematic way, even in a comprehensive survey of various relationships between orthogonal polynomials and probability theory as [8]. Excepting the Meixner- Pollaczek distribution, see e.g. [7] p.154, the use of these distributions as models remains a widely open field. Therefore the aim of the present paper is to con- tribute to filling this gap and provide basic results for further use of the fruitful Askey scheme in the field of probability and statistics. 2 J.-R. Pycke Concerning the Askey scheme, notations, definitions and results about continu- ous polynomials of the Askey scheme we deal with in the present paper, complete and self-contained references for all formulas are [3], [4]. See also Chapter 1 and Appendix E in [8] and [5]. The Askey scheme of continuous polynomials admits the graph representation Wilson (1.1) r w Continuous dual Hahn Continuous Hahn t ' Meixner-Pollaczek Jacobi ) w Laguerre $ ~ Hermite Each vertex corresponds to a parametric family of sequences of orthogonal polyno- mials named after the mathematician who introduced or studied the corresponding family. The arrows represent mathematical relations that will be made explicit below. With polynomial notations the graph becomes 2 Wk(x ; a; b; c; d) s w 2 (a;b) Sk(x ; a; b; c) Hk (x) x ' P (λ)(x; φ) P (α,β)(x) & w L(α)(x) Hk(x) Probabilistic Askey scheme 3 Arrows correspond to the limit relations 2 Wk(x ; a; b; c; d) 2 lim = Sk(x ; a; b; c) d!1 (a + d)k 2 Wk((x + t) ; a − it; b − it; a + it; b + it) lim = Hk(x; a; b) t!1 ((−2t)kk! 2 1−x α+1 α+1 β+1 β+1 Wk(t ; ; ; + it; − it) lim 2 2 2 2 2 = P (α,β)(x) t!1 k!t2k S ((x − t)2; λ + it; λ − it; t cot φ) lim k = P (λ)(x; φ) t!1 t k k!( sin φ )k Hλ+it;−t tan φ(x − t) lim k = P (λ)(x; φ) t!1 k it k i ( cos φ )k α+1+it ; β+1−it H 2 2 (− xt ) lim k 2 = P (α,β)(x) t!1 (−t)k k α+1 ( 2 ) x α lim Pk (− ; φ) = Lk (x) φ!0 2φ p x λ − λ cos φ H (x) lim λ−k=2P (λ)( ; φ) = k λ!1 k sin φ k! 2x lim P (α,β)(1 − ) = L(α)(x) β!1 k β k x H (x) lim α−k=2P (α,α)(p ) = k α!1 k α 2kk! k (α) p (−1) lim L (x 2α + α) = Hk(x): α!1 k k! Each family of orthogonal polynomials is related to a real weight-function via orthogonality relations (2.1). In view of applications in probability theory and statistics we will limit ourselves to parameters for which the weight function is non-negative. In that case after normalization the weight-function gives rise to a probability density function (p.d.f). The weight functions of the Askey-Wilson scheme and hypotheses on coefficients are the following, the name of the corre- 4 J.-R. Pycke sponding orthogonal families being given each case. 2 1 G(a + ix)G(b + ix)G(c + ix)G(d + ix) fa;b;c;d(x) = (Wilson) 2π G(2ix) x > 0; R(a; b; c; d) > 0; non-real parameters occuring in conjugate pairs; (1.2) 2 1 G(a + ix)G(b + ix)G(c + ix) fa;b;c(x) = (continuous dual Hahn) 2π G(2ix) x > 0; R(a; b; c) > 0; non-real parameters occurring in a conjugate pair; (1.3) −1 2 fa;b(x) = (2π) jG(a + ix)G(b + ix)j ; (continuous Hahn) x 2 R; R(a; b) > 0; (1.4) f (λ)(x; φ) = (2π)−1e(2φ−π)xjG(λ + ix)j2 (Meixner-Pollaczek) x 2 R; λ > 0; 0 < φ < π; (1.5) f (α,β)(x) = (1 − x)α(1 + x)β; −1 < x < 1; α; β > −1 (Jacobi); (1.6) f (α)(x) = xαe−x; x > 0; α > −1 (Laguerre) ; (1.7) −x2 f0(x) = e ; x 2 R (Hermite). (1.8) Remark 1.1. Hypotheses upon (a; b) in (1.2); (1.3); (1.4) are nothing but the same as those appearing in the definition the corresponding polynomials. Note that in (1.4) restriction in terms of conjugation as in (1.2) or (1.3) disappeared, a and b arising as limits of two pairs of conjugate coefficients, see relation (1.12). The p.d.f. corresponding to the weight-function f will be denoted by f˜. Note that whereas the weight function corresponding to a given family is subject to variations in the literature, up to a constant factor, the corresponding p.d.f. is unique. Probabilistic Askey scheme 5 Our Askey scheme for p.d.f.’s will therefore be the following. f˜a;b;c;d(x) (1.9) t y f˜a;b;c(x) f˜a;b(x) y % f˜(λ)(x; φ) f˜(α,β)(x) % y f˜(α)(x) Ó f˜0(x) In Proposition 2.1 relation (2.2) gives normalization constants as a function of one of the coefficient related to the corresponding orthogonal family. It enables us to give the explicit closed form of the p.d.f.’s for the seven families, see formulas (3.3),(4.4), (5.3),(6.2),(7.3), (8.3) and (9.3). Furthermore formulas (3.4); (4.5); (5.4); (6.2); (7.4); (8.4); (9.5) describe when necessary the asymptotic behavior of distribution tails providing as a corollary the domain of the moment generating function (m.g.f.). Concerning the latter, we will not give more detail as a closed form when it is available and the limit relations. We postpone this issue to a forthcoming paper. With obvious notations random variables with these distributions will be de- noted by Wil2(a; b; c; d); CDHahn2(a; b; c); CHahn(a; b; ); MeixPol(λ, φ); Jac(α; β); Lag(α); Her: Remark 1.2. Wilson and continuous dual Hahn polynomials are defined as poly- nomials of the variable x2. This is the reason why we consider the corresponding p.d.f as that of squared random variables. Remark 1.3. As mentioned above the random variables related to Jacobi, Laguerre and Hermite polynomials are closely related to the well-known Beta, Gamma and 6 J.-R. Pycke Gaussian densities respectively. More precisely, if these random variables and their densities are G(α + β) Beta(α; β) : xα(1 − x)β (0 < x < 1) G(α)G(β) βα Gamma(α; β) : xα−1e−βx (x > 0) G(α) 2 2 −1=2 − x N(µ, σ ) : (2π) e 2σ2 (x 2 R) it is readily checked that 1 − Jac(α − 1; β − 1) Lag(α − 1) Beta(α; β) = ; Gamma(α; β) = ; 2 β p N(µ, σ2) = µ + σ 2 Her: The Askey scheme for these random variables is the following. Wil2(a; b; c; d) (1.10) s w CDHahn2(a; b; c) CHahn(a; b) v ' MeixPol(λ, φ) Jac(α; β) ( v Lag(α) ! ~ Her It is shown in Proposition 2.2 that the sequence of moments of these random vari- ables is related to the sequence of polynomials via the elementary but far-reaching formula (2.3). In view of applications we give the first two or three polynomials of each family. They allow to compute the expectation and variance of our random variables, see formulas (3.2); (4.3); (5.2); (6.1); (7.2); (8.2); (9.4). The limit relations between p.d.f.’s corresponding to limit relations between polynomials are given by (3.5), (3.7), (3.8), (4.6), (5.5). (5.6), (6.3), (6.4), (7.5), (7.6) and (8.5). These convergences of densities, when combined with Lemma Probabilistic Askey scheme 7 2.3 which is a particular case of Scheffé’s Lemma, will imply in turn the following limit relations between our random variables.