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 Probability Surveys > Vol. 7 (2010) open journal systems 

Moments of Gamma type and the Brownian supremum process area

Svante Janson, Uppsala University

We study positive random variables whose moments can be expressed by products and quotients of Gamma functions; this includes many standard distributions. General results are given on existence, series expansion and asymptotics of density functions. It is shown that the integral of the supremum process of Brownian motion has moments of this type, as well as a related random variable occurring in the study of hashing with linear displacement, and the general results are applied to these variables.

Addendum: An addendum is published in Probability Surveys 7 (2010) 207–208.

AMS 2000 subject classifications: Primary 60E10; secondary 60J15.

Keywords: Moments, Gamma function, Brownian motion, supremum process, generalized Pólya urns.

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Janson, Svante, Moments of Gamma type and the Brownian supremum process area, Probability Surveys, 7, (2010), 1-52 (electronic). DOI: 10.1214/10-PS160.


[1]    M. Abramowitz & I. A. Stegun, eds., Handbook of Mathematical Functions. Dover, New York, 1972.

[2]     C. Berg, The cube of a normal distribution is indeterminate. Ann. Probab. 16 (1988), no. 2, 910–913. MR0929086

[3]    N. H. Bingham, C. M. Goldie & J. L. Teugels, Regular Variation. Cambridge Univ. Press, Cambridge, 1987. MR0898871

[4]    R. Blumenfeld & B. B. Mandelbrot, Lévy dusts, Mittag-Leffler statistics, mass fractal lacunarity, and perceived dimension. Phys. Rev. E (3) 56 (1997), no. 1, part A, 112–118. MR1459089

[5]    L. Bondesson, Generalized gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics 76, Springer-Verlag, New York, 1992. MR1224674

[6]    G. Doetsch, Handbuch der Laplace-transformation I, Birkhäuser, Basel, 1950. MR0344808

[7]     F. Eggenberger & G. Pólya, Über die Statistik verketteter Vorgänge. Zeitschrift Angew. Math. Mech. 3 (1923), 279–289.

[8]    W. Feller, Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67 (1949), 98–119. MR0032114

[9]    W. Feller, An Introduction to Probability Theory and its Applications, Volume II, 2nd ed., Wiley, New York, 1971. MR0270403

[10]    P. Flajolet, P. Dumas & V. Puyhaubert, Some exactly solvable models of urn process theory. Proc. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, Discrete Math. Theor. Comput. Sci. Proc. AG, 2006, 59–118. MR2509623

[11]    P. Flajolet, X. Gourdon & P. Dumas, Mellin transforms and asymptotics: harmonic sums. Theor. Computer Science 144 (1995), 3–58. MR1337752

[12]    P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, Cambridge, UK, 2009. MR2483235

[13]    R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics. 2nd ed., Addison–Wesley, Reading, Mass., 1994. MR1397498

[14]     A. Gut, Probability: A Graduate Course. Springer, New York, 2005. MR2125120

[15]    S. Janson, Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 (2004), no. 2, 177–245. MR2040966

[16]    S. Janson, Limit theorems for triangular urn schemes. Probab. Theory Rel. Fields 134 (2005), 417–452. MR2226887

[17]    S. Janson, Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas. Probability Surveys 4 (2007), 80–145. MR2318402

[18]     S. Janson & G. Louchard, Tail estimates for the Brownian excursion area and other Brownian areas. Electronic J. Probab. 12 (2007), no. 58, 1600–1632. MR2365879

[19]    S. Janson & N. Petersson, The integral of the supremum process of Brownian motion. J. Appl. Probab. 46 (2009), no. 2, 593–600. MR2535835

[20]    N. N. Lebedev, Special Functions and their Applications. (Translated from Russian.) Dover, New York, 1972. MR0350075

[21]    M. R. Leadbetter, G. Lindgren & H. Rootzén, Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York, 1983. MR0691492

[22]    J. Marcinkiewicz, Sur les fonctions indépendants III. Fund. Math. 31 (1938), 86–102.

[23]    G. Mittag-Leffler, Sur la représentation analytique d’une branche uniforme d’une fonction monogène (cinquième note). Acta Math. 29 (1905), no. 1, 101–181. MR1555012

[24]    N. Petersson, The maximum displacement for linear probing hashing I. Tech. report 2008:8, Uppsala. In The Maximum Displacement for Linear Probing Hashing, Ph. D. thesis, Uppsala, 2009.

[25]    R. N. Pillai, On Mittag-Leffler functions and related distributions. Ann. Inst. Statist. Math. 42 (1990), no. 1, 157–161. MR1054728

[26]    H. Pollard, The completely monotonic character of the Mittag-Leffler function Ea(-x). Bull. Amer. Math. Soc. 54 (1948), 1115–1116. MR0027375

[27]     G. Pólya, Sur quelques points de la théorie des probabilités. Ann. Inst. Poincaré 1 (1931), 117–161.

[28]    P. Protter, Stochastic Integration and Differential Equations. 2nd ed., Springer, Berlin, 2004. MR2020294

[29]    V. Puyhaubert, Modèles d’urnes et phénomènes de seuils en combinatoire analytique. Ph. D. thesis, École Polytechnique, Palaiseau, 2005.

[30]    D. Revuz & M. Yor, Continuous Martingales and Brownian Motion. 3rd ed., Springer–Verlag, Berlin, 1999. MR1725357

[31]     E. V. Slud, The moment problem for polynomial forms in normal random variables. Ann. Probab. 21 (1993), no. 4, 2200–2214. MR1245307

[32]     T. J. Stieltjes, Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse Math. 8(J) (1894), 1–122; 9(A) (1895), 1–47. Reprinted in Oeuvres Complètes 2, pp. 402–566, Noordhoff, Groningen, 1918. MR1344720

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Probability Surveys. ISSN: 1549-5787