On a generalized Gamma convolution related to the q-calculus

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1 On a generalized Gamma convolution related to the q-calculus Christian Berg May 22, 23 Abstract We discuss a probability distribution I q depending on a parameter < q < and determined by its moments n!/(q; q) n. The treatment is purely analytical. The distribution has been discussed recently by Bertoin, Biane and Yor in connection with a study of exponential functionals of Lévy processes. 2 Mathematics Subject Classification: primary 33D5; secondary 6E7. Keywords: q-calculus, infinitely divisible distribution Introduction In [3] Bertoin et al. studied the distribution I q of the exponential functional I q = q Nt dt, () where < q < is fixed and (N t, t ) is a standard Poisson process. They found the density i q (x), x > and its Laplace and Mellin transforms. They also showed that a simple construction from I q leads to the density λ q (x) = log(/q)(q, x, q/x; q), (2) found by Askey, cf. [2], and having log-normal moments. The notation in (2) is the standard notation from [2], see below. The distribution I q has also appeared in recent work of Cowan and Chiu [9], Dumas et al. [2] and Pakes [24]. The proofs in [3] rely on earlier work on exponential functionals which use quite involved notions from the theory of stochastic processes, see [7],[8].

2 The purpose of this note is to give a self-contained analytic treatment of the distribution I q and its properties. In Section 2 we define a convolution semigroup (I q,t ) t> of probabilities supported by [, [, and it is given in terms of the corresponding Bernstein function f(s) = log( s; q) with Lévy measure ν on ], [ having the density dν dx = x exp( xq n ). (3) n= The function / log( s; q) is a Stieltjes transform of a positive measure which is given explicitly, and this permits us to determine the potential kernel of (I q,t ) t>. The measure I q := I q, is a generalized Gamma convolution in the sense of Thorin, cf. [27], [28]. The moment sequence of I q is shown to be n!/(q; q) n, and the n th moment of I q,t is a polynomial of degree n in t. We give a recursion formula for the coefficients of these polynomials. We establish that I q has the density i q (x) = exp( xq n ) ( )n q n(n )/2. (q; q) n (q; q) n= A treatment of the theory of generalized Gamma convolutions can be found in Bondesson s monograph [6]. The recent paper [5] contains several examples of generalized Gamma convolutions which are also distributions of exponential functionals of Lévy processes. We shall use the notation and terminology from the theory of basic hypergeometric functions for which we refer the reader to the monograph by Gasper and Rahman [2]. We recall the q-shifted factorials n (z; q) n = ( zq k ), z C, < q <, n =, 2,..., k= and (z; q) =. Note that (z; q) is an entire function of z. For finitely many complex numbers z, z 2,..., z p we use the abbreviation (z, z 2,..., z p ; q) n = (z ; q) n (z 2 ; q) n... (z p ; q) n. The q-shifted factorial is defined for arbitrary complex index λ by (z; q) λ = (z; q) (zq λ ; q), and this is related to Jackson s function Γ q defined by Γ q (z) = (q; q) z ( q) z = (q; q) (q z ; q) ( q) z. (4) 2

3 In Section 3 we introduce the entire function h(z) = Γ(z)(q z ; q) and use it to express the Mellin transform of I q. We finally show that the density λ q given in (2) can be written as the product convolution of I q and another related distribution, see Theorem 3.2 below. The Mellin transform of the density λ q can be evaluated as a special case of the Askey-Roy beta-integral given in [4] and in particular we have, see also [3]: t c dt ( t, q/t; q) t = (q; q) Γ(c)Γ( c) ( q), c C \ Z. (5) Γ q (c)γ q ( c) The value of (5) is an entire function of c and equals h(c)h( c)/(q; q). The following formulas about the q-exponential functions, cf. [2], are important in the following: e q (z) = n= z n (q; q) n = (z; q), z <, (6) E q (z) = n= q n(n )/2 z n (q; q) n = ( z; q), z C. (7) 2 The analytic method We recall that a function ϕ : ], [ [, [ is called completely monotonic, if it is C and ( ) k ϕ (k) (s) for s >, k =,,.... By the Theorem of Bernstein completely monotonic functions have the form ϕ(s) = e sx dα(x), (8) where α a non-negative measure on [, [. Clearly ϕ(+) = α([, [). The equation (8) expresses that ϕ is the Laplace transform of the measure α. To establish that a probability η on [, [ is infinitely divisible, one shall prove that its Laplace transform can be written e sx dη(x) = exp( f(s)), s, where the non-negative function f has a completely monotonic derivative. If η is infinitely divisible, there exists a convolution semigroup (η t ) t> of probabilities on [, [ such that η = η and it is uniquely determined by e sx dη t (x) = e tf(s), s >, 3

4 cf. [],[2]. The function f is called the Laplace exponent or Bernstein function of the semigroup. It has the integral representation f(s) = as + ( e sx ) dν(x), (9) where a and the Lévy measure ν on ], [ satisfies the integrability condition x/( + x) dν(x) <. If f is not identically zero the convolution semigroup is transient with potential kernel κ = η t dt, and the Laplace transform of κ is /f since e sx dκ(x) = ( e sx dη t (x))dt = e tf(s) dt = f(s). () The generalized Gamma convolutions η are characterized among the infinitely divisible distributions by the following property of the corresponding Bernstein function f, namely by f being a Stieltjes transform, i.e. of the form f (s) = a + dµ(x) s + x, s >, where a and µ is a non-negative measure on [, [. The relation between µ and ν is that dν dx = e xy dµ(y). x This result was used in [6] to simplify the proof of a theorem of Thorin ([27]), stating that the Pareto distribution is a generalized Gamma convolution. Theorem 2. Let < q < be fixed. The function f(s) = log( s; q) = log( + sq n ), s () n= is a Bernstein function. The corresponding convolution semigroup ((I q,t ) t> ) consists of generalized Gamma convolutions and we have e sx di q,t (x) = e tf(s) =, s >. (2) ( s; q) t The potential kernel κ q = I q,t dt has the following completely monotonic density k q (x) = q + e xy ϕ(y) dy, (3) where ϕ is the continuous function { ( n log ϕ(x) = 2 (x; q) + n 2 π 2) if q (n ) < x < q n, n =, 2,... if x = q n, n =,,.... (4) 4

5 Proof: The function f defined by () has the derivative f (s) = n= s + q n, showing that f is a Stieltjes transform with a =, and µ is the discrete measure with mass in each of the points q n, n. In particular f is a Bernstein function with a = and Lévy measure given by (3). Since log( + s) dx = s x + s x we get ([x] denoting the integer part of x) f(s) s = [log x/ log(/q)] + (x + s)x showing that f(s)/s is a Stieltjes transform. It follows by the Reuter-Itô Theorem, cf.[22],[25],[5], that /f(s) is a Stieltjes transform. Since f is an increasing function mapping ], [ onto the real line with f() = and f () = /( q) we get f(s) = q dµ(x) + s x + s, where dx dµ(x) = lim y + π Im{ f( x + iy) } dx in the vague topology. For x ] q (n ), q n[, n =, 2,... we find lim y + f( x + iy) = ( n log q k x + inπ + k= k=n = (log (x; q) + inπ). log( q k x) ) These expressions define in fact a continuous function on [, [, vanishing at the points q n, n, so the measure µ has the density ϕ given by (4). Using that the Stieltjes transformation is the second iteration of the Laplace transformation, the assertion about the potential kernel κ q follows. Denoting by E a, a > the exponential distribution with density a exp( ax) on the positive half-line, we have e sx de a (x) = e log(+s/a), s, so we can write I q := I q, as the infinite convolution I q = n= E q n. 5

6 If we let Γ a,t denote the Gamma distribution with density then we similarly have x Specializing (2) to t = we have at Γ(t) xt e ax, x >, I q,t = n= Γ q n,t. e sx di q (x) = ( s; q), s >, (5) and since the right-hand side of (5) is meromorphic in C with poles at s = q n, n and in particular holomorphic for s <, we know that I q has moments of any order with s n (I q ) = ( ) n D n { } s=, n =,,..., ( s; q) cf. [23, p.36]. Here and in the following we denote by s n (µ) the n th moment of the measure µ. However by (6) we have hence ( s; q) = n= ( s) n (q; q) n, (6) s n (I q ) = n! (q; q) n. (7) Since I q has an analytic characteristic function, the corresponding Hamburger moment problem is determinate. By Stirling s formula we have n= 2n s 2n (I q ) =, so also Carleman s criterion shows the determinacy, cf. []. By [7, Cor. 3.3] follows that I q,t is determinate for all t > and by [9] the n th moment s n (I q,t ) is a polynomial of degree n in t given by s n (I q,t ) = n k= where the coefficients c n,k satisfy the recurrence equation n ( ) n c n+,l+ = c k,l σ n k. k k=l 6 c n,k t k, n, (8)

7 Here σ n = ( ) n f (n+) (), where f is given by (), cf. [9, Prop. 2.4], so σ n is easily calculated to be n! σ n =, n. qn+ It follows also by [9] that c n,n = σ n = ( q) n, c n, = σ n = (n )!/( q n ). Defining d n,k = ( q) k c n,k we have and s n (I q,t ) = d n+,l+ = n! n k=l n k= d k,l k! ( ) k t d n,k, n, (9) q ( n k ) q j, l =,,..., n. (2) j= In particular d n,n =, d n,n = ( n 2) + q, d n, = (n )! ( n ) q j. j= We give the first coefficients d, = d 2,2 =, d 2, = + q d 3,3 =, d 3,2 = 3 + q, d 3, = 2 + q + q 2. It follows by induction using (2) that d n,k as a function of q has a finite limit for q. The image measures µ t = τ q (I q,t ) under τ q (x) = x( q) form a convolution semigroup (µ t ) t> with e sx dµ t (x) =, s >, ( s( q); q) t and s n (µ t ) = n d n,k ( q) n k t k. k= 7

8 It follows that s n (µ t ) t n for q, so lim q µ t = δ t weakly by the method of moments. This is also in accordance with lim q ( s( q); q) t = e st, because the q-exponential function E q given in (7) converges to the exponential function in the following sense cf. [2]. lim E q(z( q)) = exp(z), q Remark 2.2 Consider a non-zero Bernstein function f. proved by probabilistic methods that the sequence In [7], [8] it was s n = n! f()... f(n) is a determinate Stieltjes moment sequence, meaning that it is the moment sequence of a unique probability on [, [. The special case f(s) = q s gives the moment sequence (7). In [] the above result of [7], [8] is obtained as a special case of the following result: Let (a n ) be a non-vanishing Hausdorff moment sequence. Then (s n ) defined by s = and s n = /(a... a n ) for n is a normalized Stieltjes moment sequence. In order to find an expression for I q we consider the discrete signed measure µ q = k= ( ) k q k(k+)/2 (q; q) k (q; q) δ q k (2) with moments s n (µ q ) = k= ( ) k q nk q k(k+)/2 (q; q) k (q; q) = (qn+ ; q) (q; q) = (q; q) n, where we have used (7). In particular, the signed measure µ q has mass. For measures ν, τ on ], [ we define the product convolution ν τ as the image of the product measure ν τ under x, y xy. The product convolution is the ordinary convolution of measures on the locally compact abelian group ], [ with multiplication as group operation. In particular we have f dν τ = f(xy) dν(x) dτ(y). 8

9 From this equation we get the moment equation s n (ν τ) = s n (ν)s n (τ), hence s n (µ q E ) = n!/(q; q) n, which shows that µ q E has the same moments as I q. Since the first measure is not known to be non-negative, we cannot conclude right-away that the two measures are equal, although I q is Stieltjes determinate. We shall show that µ q E has a density i q (x), which is non-negative. Since δ a E = E /a for a >, it is easy to see that i q (x) = n= but it is not obvious that i q (x). exp( xq n ) ( )n q n(n )/2 (q; q) n (q; q), (22) Proposition 2.3 The function i q (x) given by (22) is non-negative for x >. Therefore I q = µ q E = i q (x) ], [ (x) dx. Proof: The Laplace transform of the function i q is n= ( ) n q n(n )/2 (s + q n )(q; q) n (q; q), (23) which is the partial fraction expansion of /( s; q), since the residue of this function at the pole s = q n is We claim that ( s; q) = n= ( ) n q n(n )/2 (q; q) n (q; q). ( ) n q n(n )/2 (s + q n )(q; q) n (q; q), s q n, n =,,..., (24) which shows that I q and i q have the same Laplace transform, so i q is the density of I q and hence non-negative. To see the equation (24) we note that the left-hand side minus the right-hand side of the equation is an entire function φ, and by (6) we get φ (n) () n! = ( )n (q; q) n ( )n (q; q) k= q (n+)k ( )k q k(k )/2 (q; q) k, but by (7) the sum above equals (q n+ ; q), and we get φ (n) ()/n! =, which shows that φ is identically zero. 9

10 We call the attention to the fact that the identity (24) was also used in the work [2] of Dumas et al., but it is in fact a special case of Jackson s transformations, see (III 4) in [2] with b =, a = s, z = q. Let R q denote the following positive discrete measure with moments R q = (q; q) s n (R q ) = (q; q) k= k= q k (q; q) k δ q k (25) (q n+ ) k (q; q) k = (q; q) n, (26) by (6). We claim that µ q given by (2) and R q are the inverse of each other under the product convolution, i.e. δ = µ q R q. (27) This amounts to proving that n ( ) k q k(k+)/2 q n k = δ n, n, (q; q) k (q; q) n k k= but this follows by Cauchy multiplication of the power series (6), (7). Combining Proposition 2.3 with (27) we get: Corollary 2.4 The following factorization hold E = I q R q, which corresponds to the factorization of the moments of E as n! = n! (q; q) n (q; q) n. Remark 2.5 The factorization of Corollary 2.4 is a special case of a general factorization in [4]: E = I f R f, n! = n! (f()... f(n)), f()... f(n) where f is a non-zero Bernstein function (9), and I f, R f are determined by their moments s n (I f ) = n! f()... f(n), s n(r f ) = f()... f(n).

11 3 The entire function h(z) := Γ(z)(q z ; q) Since the Gamma function has simple poles at z = n, n =,,..., where (q z ; q) has simple zeros, it is clear that the product h(z) := Γ(z)(q z ; q) is entire. We have h() = lim z Γ(z)(q z ; q) = lim z Γ(z + )(q z+ ; q) q z z and from this it is easy to see that = (q; q) log(/q), Proposition 3. For z C we have n= h( n) = (q; q) n q n(n+)/2 (q; q) log(/q). (28) n! x z di q (x) = h(z + ) (q; q). (29) Proof: For Rz > the following calculation holds by (22) and (7): x z ( ) n q n(n )/2 di q (x) = x z e xq n dx = (q; q) Γ(z + )(q z+ ; q). (q; q) n (q; q) Since the right-hand side is entire and I q is a positive measure, we get by a classical result (going back to Landau for Dirichlet series, see [29, p.58]) that the integral in (29) must converge for all z C, and therefore the equation holds for all z C. When discussing measures on ], [ it is useful to consider this set as a locally compact group under multiplication. The Haar measure is then dm(x) = (/x) dx, and it is useful to consider the density of a measure with respect to the Haar measure m. The Mellin transformation is the Fourier tranformation of the locally compact abelian group (], [, ), and when the dual group is realized as the additive group R, the Mellin transformation of a finite measure µ on ], [ is defined as We get from (29) that M(µ)(ξ) = M(I q )(ξ) = x iξ dµ(x), ξ R. h( iξ) (q; q). (3) From Proposition 3. it follows that I q has negative moments of any order, and from (28) we get in particular that J q := x log(/q) di q(x) (3)

12 is a probability. The image of J q under the reflection x /x is denoted ˇJ q. Theorem 3.2 The product convolution L q := I q ˇJ q has the density (2) λ q (x) = log(/q)(q, x, q/x; q) with respect to Lebesgue measure on the half-line. Proof: For z C we clearly have and by (29) we get x z dl q (x) = By (5) it follows that for z C x z dl q (x) = x z dl q (x) = x z di q (x) h(z + )h( z). log(/q)(q; q) 2 x z λ q (x) dx, x z dj q (x), so L q = λ q (x) dx. Remark 3.3 In [3] the authors prove Theorem 3.2 by showing that ( x z ) dl q (x) = x z ( ) n q n(n+)/2 dx (q; q) 3 log(/q) x + q n n= for < Rz <, and then they prove the partial fraction expansion of the meromorphic density λ q (x) λ q (x) = (q; q) 3 log(/q) n= Remark 3.4 The moments of ˇJ q are given by ( ) n q n(n+)/2 x + q n. so s n ( ˇJ q ) = (q; q) n q (n+)n/2, (32) n! n= 2n s n ( ˇJ <. q ) 2

13 Therefore Carleman s criterion gives no information about determinacy of ˇJ q. By the Krein criterion, cf. [8],[26], we can conclude that log i q (x) x( + x) dx =, because I q is determinate. The substitution x = /y in this integral leads to but since log i q (/y) y( + y) dy =, (33) ˇJ q = i q(/y) dy y log(/q), we see that (33) gives no information about indeterminacy of ˇJq. We do not know if ˇJ q is determinate or indeterminate, and as a factor of an indeterminate distribution L q none of these possibilities can be excluded. References [] N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis. English translation, Oliver and Boyd, Edinburgh, 965. [2] R. Askey, Limits of some q-laguerre polynomials, J. Approx. Theory 46 (986), [3] R. Askey, Orthogonal polynomials and theta functions. In: Theta functions Bowdoin 987, ed. L. Ehrenpreis, R. C. Gunning. Proc. Symp. Pure Math. 49-part 2 (989), [4] R. Askey and R. Roy, More q-beta integrals, Rocky Mountain J. Math. 6 (986), [5] C. Berg Quelques remarques sur le cône de Stieltjes. In: Séminaire de Théorie du Potentiel Paris, No. 5. Lecture Notes in Mathematics 84. Springer-Verlag, Berlin-Heidelberg-New York, 98. [6] C. Berg, The Pareto distribution is a generalized Γ-convolution. A new proof, Scand. Actuarial J. (98), 7 9. [7] C. Berg, On the preservation of determinacy under convolution, Proc. Amer. Math. Soc. 93 (985), [8] C. Berg, Indeterminate moment problems and the theory of entire functions, J. Comp. Appl. Math. 65 (995),

14 [9] C. Berg, On infinitely divisible solutions to indeterminate moment problems, pp. 3 4 in: Proceedings of the International Workshop Special Functions, Hong Kong, 2-25 June, 999. Ed. C. Dunkl, M. Ismail, R. Wong. World Scientific, Singapore 2. [] C. Berg, A. J. Duran, A transformation from Hausdorff to Stieltjes moment sequences, Preprint Series 22, No. 9, Department of Mathematics, University of Copenhagen. [] C. Berg, G. Forst, Potential theory on locally compact abelian groups, Springer-Verlag, Berlin-Heidelberg-New York, 975. [2] J. Bertoin, Lévy processes. Cambridge University Press, Cambridge, 996. [3] J. Bertoin, P. Biane and M. Yor, Poissonian exponential functionals, q- series, q-integrals, and the moment problem for the log-normal distribution. To appear in the Proceedings of the Ascona Conference, May 22, Birkhäuser. [4] J. Bertoin, M. Yor, On subordinators, self-similar Markov processes and some factorizations of the exponential variable, Elect. Comm. in Probab. 6 (2) [5] P. Biane, J. Pitman and M. Yor, Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. Amer. Math. Soc. 38 (2), [6] L. Bondesson, Generalized gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics 76, Springer, New York, 992. [7] P. Carmona, F. Petit and M. Yor, Sur les fonctionelles exponentielles de certains processus de Lévy, Stochastics and Stochastics Reports 47 (994), 7. [8] P. Carmona, F. Petit and M. Yor, On the distribution and asymptotic results for exponential functionals of Levy processes. In: M. Yor (editor),exponential functionals and principal values related to Brownian motion, pp Biblioteca de la Revista Matemática Iberoamericana. [9] R. Cowan, S-.N. Chiu, A stochastic model of fragment formation when DNA replicates, J. Appl. Probab. 3, (994), [2] V. Dumas, F. Guillemin and Ph. Robert, A Markovian analysis of additiveincrease multiplicative-decrease algorithms, Adv. in Appl. Probab. 34 (22), 85. 4

15 [2] G. Gasper and M. Rahman, Basic Hypergeometric Series. Cambridge University Press, Cambridge, 99. [22] M. Itô, Sur les cônes convexes de Riesz et les noyaux complètement sousharmoniques, Nagoya Math. J. 55 (974), 44. [23] E. Lukacs, Characteristic functions, Griffin, London 96. [24] A.G. Pakes, Length biasing and laws equivalent to the log-normal, J. Math. Anal. Appl. 97 (996), [25] G.E.H. Reuter, Uber eine Volterrasche Integralgleichung mit totalmonotonem Kern, Arch. Math. 7 (956), [26] J. Stoyanov, Krein condition in probabilistic moment problems, Bernoulli 6 (2), no. 5, [27] O. Thorin, On the infinite divisibility of the Pareto distribution, Scand. Actuarial. J. (977), 3 4. [28] O. Thorin, On the infinite divisibility of the lognormal distribution, Scand. Actuarial. J. (977), [29] D.V. Widder, The Laplace Transform. Princeton University Press, Princeton, 94. C. Berg, Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2, Denmark; bergmath.ku.dk 5