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Moments of Gamma type and the Brownian supremum process area
Svante Janson
Source: Probab. Surveys Volume 7
(2010), 1-52.
Abstract
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.
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Permanent link to this document: http://projecteuclid.org/euclid.ps/1272029738
Digital Object Identifier: doi:10.1214/10-PS160
Mathematical Reviews number (MathSciNet): MR2645216
Zentralblatt MATH identifier: 1194.60019
References
[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.
Mathematical Reviews (MathSciNet): MR929086
Zentralblatt MATH: 0645.60018
Digital Object Identifier: doi:10.1214/aop/1176991795
Project Euclid: euclid.aop/1176991795
[3] N. H. Bingham, C. M. Goldie & J. L. Teugels, Regular Variation. Cambridge Univ. Press, Cambridge, 1987.
Mathematical Reviews (MathSciNet): MR898871
[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.
Mathematical Reviews (MathSciNet): MR1459089
Digital Object Identifier: doi:10.1103/PhysRevE.56.112
[5] L. Bondesson, Generalized gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics 76, Springer-Verlag, New York, 1992.
Mathematical Reviews (MathSciNet): MR1224674
[6] G. Doetsch, Handbuch der Laplace-transformation I, Birkhäuser, Basel, 1950.
Mathematical Reviews (MathSciNet): MR344808
[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.
[9] W. Feller, An Introduction to Probability Theory and its Applications, Volume II, 2nd ed., Wiley, New York, 1971.
Mathematical Reviews (MathSciNet): MR270403
[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.
Mathematical Reviews (MathSciNet): MR2509623
Zentralblatt MATH: 1193.60011
[11] P. Flajolet, X. Gourdon & P. Dumas, Mellin transforms and asymptotics: harmonic sums. Theor. Computer Science 144 (1995), 3–58.
Mathematical Reviews (MathSciNet): MR1337752
Zentralblatt MATH: 0869.68057
Digital Object Identifier: doi:10.1016/0304-3975(95)00002-E
[12] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, Cambridge, UK, 2009.
Mathematical Reviews (MathSciNet): MR2483235
[13] R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics. 2nd ed., Addison–Wesley, Reading, Mass., 1994.
Mathematical Reviews (MathSciNet): MR1397498
[14] A. Gut, Probability: A Graduate Course. Springer, New York, 2005.
Mathematical Reviews (MathSciNet): MR2125120
[15] S. Janson, Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 (2004), no. 2, 177–245.
[16] S. Janson, Limit theorems for triangular urn schemes. Probab. Theory Rel. Fields 134 (2005), 417–452.
Mathematical Reviews (MathSciNet): MR2226887
Zentralblatt MATH: 1112.60012
Digital Object Identifier: doi:10.1007/s00440-005-0442-7
[17] S. Janson, Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas. Probability Surveys 4 (2007), 80–145.
Mathematical Reviews (MathSciNet): MR2318402
Zentralblatt MATH: 1189.60147
Project Euclid: euclid.ps/1178804352
[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.
Mathematical Reviews (MathSciNet): MR2365879
Zentralblatt MATH: 1189.60148
[19] S. Janson & N. Petersson, The integral of the supremum process of Brownian motion. J. Appl. Probab. 46 (2009), no. 2, 593–600.
Mathematical Reviews (MathSciNet): MR2535835
Zentralblatt MATH: 1171.60017
Digital Object Identifier: doi:10.1239/jap/1245676109
Project Euclid: euclid.jap/1245676109
[20] N. N. Lebedev, Special Functions and their Applications. (Translated from Russian.) Dover, New York, 1972.
Mathematical Reviews (MathSciNet): MR350075
[21] M. R. Leadbetter, G. Lindgren & H. Rootzén, Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York, 1983.
Mathematical Reviews (MathSciNet): MR691492
Zentralblatt MATH: 0518.60021
[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.
Mathematical Reviews (MathSciNet): MR1555012
Digital Object Identifier: doi:10.1007/BF02403200
[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.
Mathematical Reviews (MathSciNet): MR1054728
Zentralblatt MATH: 0714.60009
Digital Object Identifier: doi:10.1007/BF00050786
[26] H. Pollard, The completely monotonic character of the Mittag-Leffler function Ea(−x). Bull. Amer. Math. Soc. 54 (1948), 1115–1116.
Mathematical Reviews (MathSciNet): MR27375
Digital Object Identifier: doi:10.1090/S0002-9904-1948-09132-7
Project Euclid: euclid.bams/1183513319
[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.
Mathematical Reviews (MathSciNet): 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.
Mathematical Reviews (MathSciNet): MR1725357
[31] E. V. Slud, The moment problem for polynomial forms in normal random variables. Ann. Probab. 21 (1993), no. 4, 2200–2214.
Mathematical Reviews (MathSciNet): MR1245307
Zentralblatt MATH: 0788.60028
Digital Object Identifier: doi:10.1214/aop/1176989017
Project Euclid: euclid.aop/1176989017
[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.
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