Abstract
This paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory. We establish new asymptotic properties of this function under numerous assumptions on the ordering of the parameter and variable. Several representations involving well-known special functions are given. We also provide some integral representations and structural properties involving the “reduced” Wright function. Some of the properties established imply a reflection principle that connects the “reduced” Wright functions with the opposite values of the parameter and certain Bessel functions. Several asymptotic relations for both particular cases of this function are also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A.A. Borovkov and K.A. Borovkov, Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions, Cambridge University Press, Cambridge, 2008.
B.L.J. Braaksma, Asymptotic expansions and analytic continuations for a class of Barnes integrals, Compos. Math., 15:239–341, 1963.
J. Burridge, A. Kuznetsov, M. Kwasnicki, and A.E. Kyprianou, New families of subordinators with explicit transition probability semigroup, Stochastic Processes Appl., 124:3480–3495, 2014.
P. Hougaard, M.-L.T. Lee, and G.A. Whitmore, Analysis of overdispersed count data by mixtures of Poisson variables and Poisson processes, Biometrics, 53:1225–1238, 1997.
I.A. Ibragimov and Yu.V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen, 1971.
N.L. Johnson, S. Kotz, and A.W. Kemp, Univariate Discrete Distributions, 3rd ed., Wiley, Hoboken, NJ, 2005.
B. Jørgensen, The Theory of Dispersion Models, Chapman & Hall, London, 1997.
L.V. Kim and A.V. Nagaev, On the asymmetrical problem of large deviations, Theory Probab. Appl., 20:57–68, 1975.
C.C. Kokonendji, S. Dossou-Gbété, and C.G.B. Demétrio, Some discrete exponential dispersion models: Poisson–Tweedie and Hinde–Demétrio classes, SORT, 28:201–214, 2004.
J. Mikusiński, On the function whose Laplace-transform is \( {\mathrm{e}}^{-{z}^{\alpha }} \), Stud. Math., 18:191–198, 1959.
F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.
Yu.M. Orban and Yu.P. Studnev, Functions of Airy and Fresnel type as limiting laws for convolutions of functions of bounded variation, Ukr. Math. J., 25:255–262, 1973.
H.H. Panjer and G.E. Willmot, Insurance Risk Models, Society of Actuaries, Schaumberg, IL, 1992.
R.B. Paris, Exponentially small expansions of the Wright function on the Stokes lines, Lith. Math. J., 54(1):82–105, 2014.
R.B. Paris and D. Kaminski, Asymptotics and Mellin–Barnes Integrals, Cambridge University Press, Cambridge, 2001.
R.B. Paris and V. Vinogradov, Fluctuation properties of compound Poisson–Erlang Lévy processes, Commun. Stoch. Anal., 7:283–302, 2013.
R.B. Paris and V. Vinogradov, Branching particle systems and compound Poisson processes related to Pólya–Aeppli distributions, Commun. Stoch. Anal., 9:43–67, 2015.
R.B. Paris and V. Vinogradov, Refined local approximations for some exponential dispersion models comprised of Poisson–Tweedie mixtures, Commun. Stat., Theory Methods, 2016 (to be submitted for publication).
T.R. Prabhaker, A singular integral equation with a generalized Mittag–Leffler function in the kernel, Yokohama Math. J., 19:7–15, 1971.
L.J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, Cambridge, 1960.
R. Stankovié, On the function of E.M. Wright, Publ. Inst. Math., Nouv. Sér., 18:113–124, 1970.
V. Vinogradov, Refined Large Deviation Limit Theorems, Longman, Harlow, 1994.
V. Vinogradov, On approximations for two classes of Poisson mixtures, Stat. Probab. Lett., 78:358–366, 2008.
V. Vinogradov, R.B. Paris, and O.L. Yanushkevichiene, New properties and representations for members of the power-variance family. I, Lith. Math. J., 52(4):444–461, 2012.
V. Vinogradov, R.B. Paris, and O.L. Yanushkevichiene, New properties and representations for members of the power-variance family. II, Lith. Math. J., 53(1):103–120, 2013.
V. Vinogradov, R.B. Paris, and O.L. Yanushkevichiene, Erratum to “New properties and representations for members of the power-variance family. II”, Lith. Math. J., 54(2):229, 2014.
R. Wong and Y.-Q. Zhao, Smoothing of Stokes’s discontinuity for the generalized Bessel function, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 455:1381–1400, 1999.
E.M. Wright, The asymptotic expansion of the generalized Bessel function, Proc. Lond. Math. Soc. (2), 38:257–270, 1935.
E.M. Wright, The generalized Bessel function of order greater than one, Q. J. Math., Oxf. Ser., 11:36–48, 1940.
V.M. Zolotarev, One-Dimensional Stable Distributions, AMS, Providence, RI, 1986.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Lee Lorch, 1915–2014
Rights and permissions
About this article
Cite this article
Paris, R.B., Vinogradov, V. Asymptotic and structural properties of special cases of the Wright function arising in probability theory. Lith Math J 56, 377–409 (2016). https://doi.org/10.1007/s10986-016-9324-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-016-9324-1