A Two-Parameter Extension of Urbanik’s Product Convolution Semigroup

• Autorzy: Berg Christian

Identyfikatory

DOI

10.19195/0208-4147.39.2.11

• Języki publikacji: EN

Abstrakty

EN

We prove that s_n(a, b) = Γ(an + b)/Γ(b), n = 0, 1,…, is an infinitely divisible Stieltjes moment sequence for arbitrary a, b > 0. Its powers s_n (a, b)^c, c > 0, are Stieltjes determinate if and only if ac ≤ 2. The latter was conjectured in a paper by Lin (2019) in the case b = 1. We describe a product convolution semigroup τ_c(a, b), c > 0, of probability measures on the positive half-line with densities e_c (a, b) and having the moments s_n (a, b)^c. We determine the asymptotic behavior of e_c (a, b)(t) for t → 0 and for t → ∞, and the latter implies the Stieltjes indeterminacy when ac > 2. The results extend the previous work of the author and López (2015) and lead to a convolution semigroup of probability densities (g_c (a, b)(x))_c>0 on the real line. The special case (g_c (a, 1)(x))_c>0 are the convolution roots of the Gumbel distribution with scale parameter a > 0. All the densities g_c (a, b)(x) lead to determinate Hamburger moment problems.

Słowa kluczowe

EN

infinitely divisible Stieltjes moment sequence; product convolution semigroup; asymptotic approximation of integrals; Gumbel distribution

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2019
• Tom: Vol. 39, Fasc. 2
• Strony: 441--458
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Berg Christian

• Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark

Bibliografia

• [1] R. A. Askey and R. Roy, Chapter 5: Gamma Function, in: NIST Handbook of Mathematical Functions, NIST and Cambridge University Press, 2010.
• [2] C. Berg, On the preservation of determinacy under convolution, Proc. Amer. Math. Soc. 93 (2) (1985), pp. 351-357.
• [3] C. Berg, On infinitely divisible solutions to indeterminate moment problems, in: Proceedings of the International Workshop “Special Functions”, Hong Kong, June 21-25, 1999, C. F. Dunkl, M. Ismail, and R. Wong (Eds.), World Scientific, Singapore 2000, pp. 31-41.
• [4] C. Berg, On powers of Stieltjes moment sequences. I, J. Theoret. Probab. 18 (4) (2005), pp. 871-889.
• [5] C. Berg, On powers of Stieltjes moment sequences. II, J. Comput. Appl. Math. 199 (1) (2007), pp. 23-38.
• [6] C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Ergeb. Math. Grenzgeb., Band 87, Springer, New York-Heidelberg 1975.
• [7] C. Berg and J. L. López, Asymptotic behaviour of the Urbanik semigroup, J. Approx. Theory 195 (2015), pp. 109-121.
• [8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, D. Zwillinger (Ed.), eighth edition, Elsevier and Academic Press, Amsterdam 2015.
• [9] P. Hörfelt, The moment problem for some Wiener functionals: Corrections to previous proofs, J. Appl. Probab. 42 (3) (2005), pp. 851-860.
• [10] Z. J. Jurek, On relations between Urbanik and Mehler semigroups, Probab. Math. Statist. 29 (2) (2009), pp. 297-308.
• [11] G. D. Lin, On powers of the Catalan number sequence, Discrete Math. 342 (7) (2019), pp. 2139-2147.
• [12] G. D. Lin and J. Stoyanov, Moment determinacy of powers and products of nonnegative random variables, J. Theoret. Probab. 28 (4) (2015), pp. 1337-1353.
• [13] J. L. López, P. Pagola, and E. Pérez Sinusía, A systematization of the saddle point method: Application to the Airy and Hankel functions, J. Math. Anal. Appl. 354 (1) (2009), pp. 347-359.
• [14] F. W. J. Olver and L. C. Maximon, Chapter 10: Bessel Functions, in: NIST Handbook of Mathematical Functions, NIST and Cambridge University Press, 2010.
• [15] H. L. Pedersen, On Krein’s theorem for indeterminacy of the classical moment problem, J. Approx. Theory 95 (1) (1998), pp. 90-100.
• [16] R. L. Schilling, R. Song, and Z. Vondraček, Bernstein Functions: Theory and Applications, second edition, Walter de Gruyter & Co., Berlin 2012.
• [17] J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society, New York 1943.
• [18] A. V. Skorokhod, Asymptotic formulas for stable distribution laws, Dokl. Akad. Nauk SSSR (N.S.) 98 (1954), pp. 731-734.
• [19] T.-J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse, Math., Sér. 1, 8 (4) (1894), pp. J1-J122.
• [20] M. L. Targhetta, On a family of indeterminate distributions, J. Math. Anal. Appl. 147 (2) (1990), pp. 477-479.
• [21] Shu-gwei Tyan, The structure of bivariate distribution functions and their relation to Markov processes, Ph.D. Thesis, Princeton University, 1975.
• [22] K. Urbanik, Functionals on transient stochastic processes with independent increments, Studia Math. 103 (3) (1992), pp. 299-315.
• [23] V. M. Zolotarev, One-dimensional Stable Distributions, American Mathematical Society, Providence, RI, 1986.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).