## Electronic Communications in Probability

### Beta-gamma tail asymptotics

#### Abstract

We compute the tail asymptotics of the product of a beta random variable and a generalized gamma random variable which are independent and have general parameters. A special case of these asymptotics were proved and used in a recent work of Bubeck, Mossel, and Racz in order to determine the tail asymptotics of the maximum degree of the preferential attachment tree. The proof presented here is simpler and highlights why these asymptotics hold.

#### Article information

Source
Electron. Commun. Probab. Volume 20 (2015), paper no. 84, 7 pp.

Dates
Accepted: 9 November 2015
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465321011

Digital Object Identifier
doi:10.1214/ECP.v20-4545

Mathematical Reviews number (MathSciNet)
MR3434201

Zentralblatt MATH identifier
1328.60059

Subjects
Primary: 60E99: None of the above, but in this section

Rights

#### Citation

Pitman, Jim; Racz, Miklos. Beta-gamma tail asymptotics. Electron. Commun. Probab. 20 (2015), paper no. 84, 7 pp. doi:10.1214/ECP.v20-4545. https://projecteuclid.org/euclid.ecp/1465321011

#### References

• Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions, vol. 55, Dover, 1964.
• Addario-Berry, L.; Broutin, N.; Goldschmidt, C. Critical random graphs: limiting constructions and distributional properties. Electron. J. Probab. 15 (2010), no. 25, 741–775.
• Backhausz, Ãgnes. Limit distribution of degrees in random family trees. Electron. Commun. Probab. 16 (2011), 29–37.
• Berger, Noam; Borgs, Christian; Chayes, Jennifer T.; Saberi, Amin. Asymptotic behavior and distributional limits of preferential attachment graphs. Ann. Probab. 42 (2014), no. 1, 1–40.
• Bubeck, Sébastien; Mossel, Elchanan; RÃ¡cz, MiklÃ³s Z. On the influence of the seed graph in the preferential attachment model. IEEE Trans. Network Sci. Eng. 2 (2015), no. 1, 30–39.
• Dufresne, Daniel. Algebraic properties of beta and gamma distributions, and applications. Adv. in Appl. Math. 20 (1998), no. 3, 285–299.
• Dufresne, Daniel. $G$ distributions and the beta-gamma algebra. Electron. J. Probab. 15 (2010), no. 71, 2163–2199.
• Fox, Charles. The $G$ and $H$ functions as symmetrical Fourier kernels. Trans. Amer. Math. Soc. 98 1961 395–429.
• Goldschmidt, Christina; Haas, Bénédicte. A line-breaking construction of the stable trees. Electron. J. Probab. 20 (2015), no. 16, 24 pp.
• Hashorva, Enkelejd; Pakes, Anthony G. Tail asymptotics under beta random scaling. J. Math. Anal. Appl. 372 (2010), no. 2, 496–514.
• Hashorva, Enkelejd; Pakes, Anthony G.; Tang, Qihe. Asymptotics of random contractions. Insurance Math. Econom. 47 (2010), no. 3, 405–414.
• Janson, Svante. Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 (2006), no. 3, 417–452.
• Janson, Svante. Moments of gamma type and the Brownian supremum process area. Probab. Surv. 7 (2010), 1–52.
• Letemplier, Julien; Simon, Thomas. The area of a spectrally positive stable process stopped at zero. http://arxiv.org/abs/1410.0036
• MÃ³ri, TamÃ¡s F. The maximum degree of the BarabÃ¡si-Albert random tree. Combin. Probab. Comput. 14 (2005), no. 3, 339–348.
• PekÃ¶z, Erol A.; RÃ¶llin, Adrian; Ross, Nathan. Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 (2013), no. 3, 1188–1218.
• PekÃ¶z, Erol A.; RÃ¶llin, Adrian; Ross, Nathan. Joint degree distributions of preferential attachment random graphs, Preprint available at http://arxiv.org/abs/1402.4686, 2014.
• PekÃ¶z, Erol A.; RÃ¶llin, Adrian; Ross, Nathan. Generalized gamma approximation with rates for urns, walks and trees, The Annals of Probability, to appear (2015+).
• Springer, M. D.; Thompson, W. E. The distribution of products of beta, gamma and Gaussian random variables. SIAM J. Appl. Math. 18 1970 721–737.