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Asymptotic results for weighted means of random variables which converge to a Dickman distribution, and some number theoretical applications. (English) Zbl 1333.60031

Summary: This paper studies some examples of weighted means of random variables. These weighted means generalize the logarithmic means. We consider different kinds of random variables and we prove that they converge weakly to a Dickman distribution; this extends some known results in the literature. In some cases we have interesting connections with number theory. Moreover, we prove large deviation principles and, arguing as in [R. Giuliano and C. Macci, J. Math. Anal. Appl. 378, No. 2, 555–570 (2011; Zbl 1214.60007)], we illustrate how the rate function can be expressed in terms of the Hellinger distance with respect to the (weak) limit, i.e., the Dickman distribution.

MSC:

60F05 Central limit and other weak theorems
60F10 Large deviations
60F15 Strong limit theorems
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms

Citations:

Zbl 1214.60007
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[1] R. Arratia and S. Tavaré, Independent processes approximations for random combinatorial structures. Adv. Math.104 (1994) 90-154. · Zbl 0802.60008 · doi:10.1006/aima.1994.1022
[2] M. Atlagh and M. Weber, Le théorème central limite presque sûr. Exp. Math.18 (2000) 97-126. · Zbl 0959.60028
[3] I. Berkes and E. Csáki, A universal result in almost sure central limit theory. Stoch. Process Appl.94 (2001) 105-134. · Zbl 1053.60022 · doi:10.1016/S0304-4149(01)00078-3
[4] G.A. Brosamler, An almost everywhere central limit theorem. Math. Proc. Camb. Philos Soc.104 (1988) 561-574. · Zbl 0668.60029 · doi:10.1017/S0305004100065750
[5] F. Cellarosi and Y.G. Sinai, Non-Standard Limit Theorems in Number Theory. Prokhorov and Contemporary Probability Theory. Edited by A.N. Shiryaev, S.R.S. Varadhan and E.L. Presman. Springer, Heidelberg (2013) 197-213. · Zbl 1273.60024
[6] S. Cheng, L. Peng and L. Qi, Almost sure convergence in extreme value theory. Math. Nachr.190 (1998) 43-50. · Zbl 0932.60028 · doi:10.1002/mana.19981900104
[7] J.-M. De Koninck, I. Diouf and N. Doyon, On the truncated kernel function. J. Integer Seq.15 (2012) Article 12.3.2. · Zbl 1294.11163
[8] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. 2nd edition. Springer, New York (1998). · Zbl 0896.60013
[9] K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astron. Fys.22 (1930) 1-14. · JFM 56.0178.04
[10] P. Erdös and P. Turán, On some new questions on the distribution of prime numbers. Bull. Amer. Math. Soc.54 (1948) 371-378. · Zbl 0032.26902 · doi:10.1090/S0002-9904-1948-09010-3
[11] I. Fahrner, An extension of the almost sure max-limit theorem. Stat. Probab. Lett.49 (2000) 93-103. · Zbl 0964.60031 · doi:10.1016/S0167-7152(00)00036-5
[12] I. Fahrner and U. Stadtmüller, On almost sure max-limit theorems. Stat. Probab. Lett.37 (1998) 229-236. · Zbl 1246.60034 · doi:10.1016/S0167-7152(97)00121-1
[13] A. Fisher, Convex-invariant means and a pathwise central limit theorem. Adv. Math.63 (1987) 213-246. · Zbl 0627.60034 · doi:10.1016/0001-8708(87)90054-5
[14] A. Fisher, A pathwise central limit theorem for random walk. Preprint (1989).
[15] M. Ghosh, G.J. Babu and N. Mukhopadhyay, Almost sure convergence of sums of maxima and minima of positive random variables. Z. Wahrsch. Verw. Gebiete33 (1975) 49-54. · Zbl 0296.60017 · doi:10.1007/BF00539860
[16] R. Giuliano and C. Macci, Large deviation principles for sequences of logarithmically weighted means. J. Math. Anal. Appl.378 (2011) 555-570. · Zbl 1214.60007 · doi:10.1016/j.jmaa.2011.01.068
[17] D.A. Goldston, J. Pintz and C.M. Yıldırım, Primes in tuples. I. Ann. Math.170 (2009) 819-862. · Zbl 1207.11096 · doi:10.4007/annals.2009.170.819
[18] G.H. Hardy and E.M. Wright. An Introduction to the Theory of Numbers. 5th edition. The Clarendon Press, Oxford University Press, New York (1979). · Zbl 0423.10001
[19] M.K. Heck, The principle of large deviations for the almost everywhere central limit theorem. Stoch. Process. Appl.76 (1998) 61-75. · Zbl 0934.60022 · doi:10.1016/S0304-4149(98)00023-4
[20] D. Hensley, The convolution powers of the Dickman function. J. London Math. Soc.33 (1986) 395-406. · Zbl 0565.10041 · doi:10.1112/jlms/s2-33.3.395
[21] A. Hildebrand and G. Tenenbaum, Integers without large prime factors. J. Théor. Nombres Bordeaux5 (1993) 411-484. · Zbl 0797.11070 · doi:10.5802/jtnb.101
[22] S. Hörmann, On the universal a.s. central limit theorem. Acta Math. Hung.116 (2007) 377-398. · Zbl 1164.60016 · doi:10.1007/s10474-007-6070-1
[23] H.-K. Hwang and T.-H. Tsai, Quickselect and Dickman function. Combin. Probab. Comput.11 (2002) 353-371. · Zbl 1008.68044
[24] R. Kiesel and U. Stadtmüller, A large deviation principle for weighted sums of independent identically distributed random variables. J. Math. Anal. Appl.251 (2000) 929-939. · Zbl 0967.60025 · doi:10.1006/jmaa.2000.7176
[25] M.T. Lacey and W. Philipp, A note on the almost everywhere central limit theorem. Stat. Probab. Lett.9 (1990) 201-205. · Zbl 0691.60016 · doi:10.1016/0167-7152(90)90056-D
[26] L. Le Cam and G.L. Yang, Asymptotics in Statistics. Some Basic Concepts. Springer-Verlag, New York (1990). · Zbl 0719.62003
[27] P. Lévy. Sur certain processus stochastiques homogenes. Composition Math.7 (1939) 283-339. · JFM 65.1346.02
[28] M.A. Lifshits and E.S. Stankevich, On the large deviation principle for the almost sure CLT. Stat. Probab. Lett.51 (2001) 263-267. · Zbl 1043.60019 · doi:10.1016/S0167-7152(00)00154-1
[29] M. Loève, Probability Theory I, 4th edition. Springer-Verlag, New York (1977).
[30] P. March and T. Seppäläinen, Large deviations from the almost everywhere central limit theorem. J. Theoret. Probab.10 (1997) 935-965. · Zbl 0897.60031 · doi:10.1023/A:1022614700678
[31] A. Rouault, M. Yor and M. Zani, A large deviations principle related to the strong arc-sine law. J. Theoret. Probab.15 (2002) 793-815. · Zbl 1011.60006 · doi:10.1023/A:1016280117892
[32] P. Schatte, On strong versions of the central limit theorem. Math. Nachr.137 (1988) 249-256. · Zbl 0661.60031 · doi:10.1002/mana.19881370117
[33] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.)44 (2007) 1-18. · Zbl 1193.11086 · doi:10.1090/S0273-0979-06-01142-6
[34] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory. Translated from the second French edition by C. B. Thomas. Cambridge University Press, Cambridge (1995). · Zbl 0831.11001
[35] A.W. van der Vaart, Asymptotic Statistics. Cambridge University Press, New York (1998). · Zbl 0910.62001
[36] S.R.S. Varadhan, Large deviations and entropy. Entropy. Edited by A. Greven, G. Keller and G. Warnecke. Princeton University Press (2003) 199-214. · Zbl 1163.60312
[37] E. Westzynthius, Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind, Commun. Phys. Math. Helingsfors5 (1931) 1-37. · JFM 57.0186.02
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