Central Limit Theorem and the Distribution of Sequences

Paštéka, Milan

doi:10.2478/tmmp-2020-0031

Abstract

The paper deals with independent sequences with continuous asymptotic distribution functions. We construct a compact metric space with Borel probability measure. We use its properties to prove the central limit theorem for independent sequences with continuous distribution functions.

References

[1] BUCK, R. C.: The measure theoretic approach to density, Amer. J. Math. 68, 1946, 560–580.10.2307/2371785
Search in Google Scholar Back to article
[2] DRMOTA, M.—TICHY, R. F.: Sequences, Discrepancies and Applications. Springer-Verlag, Berlin, 1997.10.1007/BFb0093404
Search in Google Scholar Back to article
[3] ERDŐS, P.: On the density of some sequences of numbers II, J. London Math. Soc. 12 (1937), 7–11.10.1112/jlms/s1-12.45.7
Search in Google Scholar Back to article
[4] GREKOS, G.: On various definitions of density (survey), Tatra Mt. Math. Publ. 31 (2005), 17–27.
Search in Google Scholar Back to article
[5] GREKOS, G.: The density set (a survey), Tatra Mt. Math. Publ. 31 (2005), 103–111.
Search in Google Scholar Back to article
[6] GRABNER, P. J.—STRAUCH, O.—TICHY, R. F.: Lp-discrepancy and statistical independence of sequences, Czechoslovak Math. J. 49 (1999), no. 1, 97–110.
Search in Google Scholar Back to article
[7] GRABNER, P. J.—TICHY, R. F.: Remark on statistical independence of sequences, Math. Slovaca 44 (1994), no. 1, 91–94.
Search in Google Scholar Back to article
[8] IACO, M. R.—PAŠTÉKA, M.—TICHY R. F.: Measure density for set decompositions and uniform distribution, Rend. Circ. Math. Palermo 64 (2015), no. 2, 323–339.
Search in Google Scholar Back to article
[9] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. John Wiley and Sons, N. Y. London, Sydney, Toronto, 1974.
Search in Google Scholar Back to article
[10] NOVOSELOV, E. V.: Topological theory of polyadic numbers,Trudy Tbilis. Mat. Inst. 27 (1960), 61–69, (In Russian)
Search in Google Scholar Back to article
[11] NOVOSELOV, E. V.: New method in the probabilistic number theory,Izv.Doklady Akad. Nauk. Ser. Mat. 28 (1964), no. 2, 307–364. (In Russian)
Search in Google Scholar Back to article
[12] NARKIEWICZ, W.: Teoria Liczb, PWN, Warszawa, 1991. (In Polish)
Search in Google Scholar Back to article
[13] NIVEN, I.: Uniform distribution of sequences of integers, Trans.Amer. Math.Soc. 98 (1961), 52–61.10.1090/S0002-9947-1961-0120214-8
Search in Google Scholar Back to article
[14] PAŠTÉKA, M.: Some properties of Buck’s measure density, Math. Slovaca 42 (1992), no. 1, 15–32.
Search in Google Scholar Back to article
[15] PAŠTÉKA, M.: Remarks on one type of uniform distribution, Unif. Distrib. Theory 2 (2007), no. 1, 79–92.
Search in Google Scholar Back to article
[16] PAŠTÉKA, M.: On Four Approaches to Density. Spectrum Slovakia Vol. 3. Peter Lang, Frankfurt am Main; VEDA, Publishing House of the Slovak Academy of Sciences, Bratislava, 2014.
Search in Google Scholar Back to article
[17] PAŠTÉKA, M.: Remarks on Buck’s measure density, Tatra Mt. Math. Publ. 3 (1993), 191–200.
Search in Google Scholar Back to article
[18] PAŠTÉKA, M.: Density and Related Topics. Veda, Bratislava; Academia, Praha, 2017.
Search in Google Scholar Back to article
[19] PAŠTÉKA, M.—TICHY, R.: A note on the correlation coefficient of arithmetic functions, Acta Acad. Paedagog. Agriensis Sect. Math. (N.S.) 30 (2003), 109–114.
Search in Google Scholar Back to article
[20] PAŠTÉKA, M.—TICHY, R.: Measurable sequences. Riv.Mat.Univ. Parma (N.S.) 10 (2019), no. 1, 63–84.
Search in Google Scholar Back to article
[21] RAUZY, G.: Proprietes Statistique de Suites Arithmetiques.Le Mathématicien Vol. l5, Collection SUP, Presses Universitaires de France, Paris, 1976.
Search in Google Scholar Back to article
[22] SCHOENBERG, I.:Über die asymptotische Verteilung reeler Zahlen mod 1,Math. Z. 28 (1928), 171–199. https://link.springer.com/article/10.1007/BF01181156
Search in Google Scholar Back to article
[23] STRAUCH, O.: Distribution of Sequences: A Theory. VEDA, Bratislava; Academia, Prague, 2019.
Search in Google Scholar Back to article
[24] STRAUCH, O.—PORUBSKÝ,Š.: Distribution of Sequences a Sampler.In: Series of the Slovak Academy of Sciences. Vol. 1. Peter Lang, Frankfurt am Main, 2005.
Search in Google Scholar Back to article
[25] WEYL, H.:Über die Gleichverteilung von Zahlen mod. Eins,Math. Ann 77 (1916), 313–352.10.1007/BF01475864
Search in Google Scholar Back to article

Central Limit Theorem and the Distribution of Sequences

Abstract

Paradigm

My account