Modern Stochastics: Theory and Applications

Ben-Hamou, A., Boucheron, S., Ohannessian, M.I.: Concentration inequalities in the infinite urn scheme for occupancy counts and the missing mass, with applications. Bernoulli 23(1), 249–287 (2017). MR3556773. https://doi.org/10.3150/15-BEJ743

Bernstein, S.: The theory of probabilities. Gastehizdat Publishing House, Moscow (1946).

Boucheron, S., Lugosi, G., Bousquet, O.: Concentration inequalities. In: Summer School on Machine Learning, pp. 208–240 (2003). Springer.

Castillo, I., et al.: Pólya tree posterior distributions on densities. In: Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, vol. 53, pp. 2074–2102 (2017). Institut Henri Poincaré. MR3729648. https://doi.org/10.1214/16-AIHP784

Clark, C.E.: The pert model for the distribution of an activity time. Oper. Res. 10(3), 405–406 (1962).

Dumbgen, L.: New goodness-of-fit tests and their application to nonparametric confidence sets. Ann. Stat. 26(1), 288–314 (1998). MR1611768. https://doi.org/10.1214/aos/1030563987

Dutka, J.: The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24(1), 11–29 (1981). MR0618150. https://doi.org/10.1007/BF00327713

Elder, S.: Bayesian adaptive data analysis guarantees from subgaussianity. arXiv preprint arXiv:1611.00065 (2016).

Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press (2009). https://doi.org/10.1017/CBO9780511801655

Frankl, P., Maehara, H.: Some geometric applications of the beta distribution. Ann. Inst. Stat. Math. 42(3), 463–474 (1990). Available at: https://www.ism.ac.jp/editsec/aism/pdf/042_3_0463.pdf. MR1093963. https://doi.org/10.1007/BF00049302

Gauss, C.F.: Disquisitiones generales circa seriem infinitam (1813).

Gupta, A.K., Nadarajah, S.: Handbook of Beta Distribution and Its Applications. Statistics: A Series of Textbooks and Monographs. Taylor & Francis (2004). https://books.google.at/books?id=cVmnsxa-VzwC. MR2079703

Henzi, A., Duembgen, L.: Some new inequalities for beta distributions. arXiv preprint arXiv:2202.06718 (2022).

Hoeffding, W.: Probability inequalities for sums of bounded random variables. In: The Collected Works of Wassily Hoeffding, pp. 409–426 (1994). Springer. MR1307621

Ibrahim, A.K.: Contiguous relations for 2f1 hypergeometric series. J. Egypt. Math. Soc. 20(2), 72–78 (2012). MR3011583. https://doi.org/10.1016/j.joems.2012.08.005

Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, Volume 2 vol. 289. John wiley & sons (1995) MR1326603

Jones, M.: On fractional uniform order statistics. Stat. Probab. Lett. 58(1), 93–96 (2002). MR1913312. https://doi.org/10.1016/S0167-7152(02)00119-0

Kahane, J.-P.: Propriétés locales des fonctions à séries de fourier aléatoires. Stud. Math. 19(1), 1–25 (1960). MR0117506. https://doi.org/10.4064/sm-19-1-1-25

Kim, B.-c., Reinschmidt, K.F.: Probabilistic forecasting of project duration using bayesian inference and the beta distribution. J. Constr. Eng. Manage. 135(3), 178–186 (2009).

Kipping, D.M.: Parametrizing the exoplanet eccentricity distribution with the beta distribution. Mon. Not. R. Astron. Soc. Lett. 434(1), 51–55 (2013).

Marchal, O., Arbel, J., et al.: On the sub-gaussianity of the beta and dirichlet distributions. Electron. Commun. Probab. 22 (2017). MR3718704. https://doi.org/10.1214/17-ECP92

Maurer, A., Pontil, M.: Concentration inequalities under sub-gaussian and sub-exponential conditions. In: Advances in Neural Information Processing Systems 34 (2021).

Meurer, A., Smith, C.P., Paprocki, M., Čertík, O., Kirpichev, S.B., Rocklin, M., Kumar, A., Ivanov, S., Moore, J.K., Singh, S., Rathnayake, T., Vig, S., Granger, B.E., Muller, R.P., Bonazzi, F., Gupta, H., Vats, S., Johansson, F., Pedregosa, F., Curry, M.J., Terrel, A.R., Roučka, v., Saboo, A., Fernando, I., Kulal, S., Cimrman, R., Scopatz, A.: Sympy: symbolic computing in python. PeerJ Comput. Sci. 3, 103 (2017). https://doi.org/10.7717/peerj-cs.103

Mitov, K., Nadarajah, S.: Beta distributions in stochastic processes. In: Statistics Textbooks and Monographs 174, pp. 165–202 (2004). MR2079709

Mühlbach, G.v.: Rekursionsformeln für die zentralen momente der pólya-und der beta-verteilung. Metrika 19(1), 171–177 (1972). MR0375563. https://doi.org/10.1007/BF01893292

Perry, A., Wein, A.S., Bandeira, A.S., et al.: Statistical limits of spiked tensor models. In: Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, vol. 56, pp. 230–264 (2020). Institut Henri Poincaré. MR4058987. https://doi.org/10.1214/19-AIHP960

Skórski, M.: Bernstein Bounds for Beta Distribution. OSF (2023). https://doi.org/10.17605/OSF.IO/YSVDH

Stucchio, C.: Bayesian a/b testing at vwo. Whitepaper, Visual Website Optimizer (2015).

Vidūnas, R.: Contiguous relations of hypergeometric series. J. Comput. Appl. Math. 153(1-2), 507–519 (2003). MR1985719. https://doi.org/10.1016/S0377-0427(02)00643-X

Wainwright, M.J.: High-dimensional Statistics: A Non-asymptotic Viewpoint vol. 48. Cambridge University Press (2019) MR3967104. https://doi.org/10.1017/9781108627771

Williams, D.: 394: The analysis of binary responses from toxicological experiments involving reproduction and teratogenicity. Biometrics 31(4), 949–952 (1975).

Wolfram, M.: www.functions.wolfram.com/HypergeometricFunctions (2020).

Zhang, A.R., Zhou, Y.: On the non-asymptotic and sharp lower tail bounds of random variables. Stat 9(1), 314 (2020). MR4193419. https://doi.org/10.1002/sta4.314

Zhang, J., Wu, Y.: Beta approximation to the distribution of Kolmogorov-Smirnov statistic. Ann. Inst. Stat. Math. 54(3), 577–584 (2002). MR1932402. https://doi.org/10.1023/A:1022463111224