Abstract
In this paper the ideas of three types of statistical convergence of a sequence of random variables, namely, statistical convergence in probability, statistical convergence in mean of order r and statistical convergence in distribution are introduced and the interrelation among them is investigated. Also their certain basic properties are studied.
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M. Balcerzak, K. Dems, A. Komisarski: Statistical convergence and ideal convergence for sequences of functions. J. Math. Anal. Appl. 328 (2007), 715–729.
L.A. Bunimovich, Ya.G. Sinai: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78 (1981), 479–497.
P. Erdős, G. Tenenbaum: On densities of certain sequences of integers. Proc. Lond. Math. Soc., III. Ser. 59 (1989), 417–438. (In French.)
H. Fast: Sur la convergence statistique. Colloq. Math. 2 (1951), 241–244. (In French.)
J.A. Fridy: On statistical convergence. Analysis 5 (1985), 301–313.
J.A. Fridy: Statistical limit points. Proc. Am. Math. Soc. 118 (1993), 1187–1192.
J.A. Fridy, M.K. Khan: Tauberian theorems via statistical convergence. J. Math. Anal. Appl. 228 (1998), 73–95.
J.A. Fridy, C. Orhan: Statistical limit superior and limit inferior. Proc. Am. Math. Soc. 125 (1997), 3625–3631.
A.D. Gadjiev, C. Orhan: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32 (2002), 129–138.
M. Katětov: Products of filters. Commentat. Math. Univ. Carol. 9 (1968), 173–189.
E. Kolk: The statistical convergence in Banach spaces. Tartu Ül. Toimetised 928 (1991), 41–52.
P. Kostyrko, M. Macaj, T. Šalát, O. Strauch: On statistical limit points. Proc. Am.Math. Soc. 129 (2001), 2647–2654.
I. J. Maddox: Statistical convergence in a locally convex space. Math. Proc. Camb. Philos. Soc. 104 (1988), 141–145.
V.G. Martinez, G. S. Torrubia, C.T. Blanc: A statistical convergence application for the Hopfield networks. Information Theory and Applications 15 (2008), 84–88.
H. I. Miller: A measure theoretical subsequence characterization of statistical convergence. Trans. Am. Math. Soc. 347 (1995), 1811–1819.
S. Pehlivan, M.A. Mamedov: Statistical cluster points and turnpike. Optimization 48 (2000), 93–106.
M.D. Penrose, J. E. Yukich: Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003), 277–303.
V.K. Rohatgi: An Introduction to Probability Theory and Mathematical Statistics. Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 1976.
T. Šalát: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139–150.
E. Savaš: On statistically convergent sequences of fuzzy numbers. Inf. Sci. 137 (2001), 277–282.
I. J. Schoenberg: The integrability of certain functions and related summability methods. Am. Math. Mon. 66 (1959), 361–375.
H. Steinhaus: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2 (1951), 73–74.
A. Zygmund: Trigonometric Series. Cambridge University Press, 1979.
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Ghosal, S. Statistical convergence of a sequence of random variables and limit theorems. Appl Math 58, 423–437 (2013). https://doi.org/10.1007/s10492-013-0021-7
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DOI: https://doi.org/10.1007/s10492-013-0021-7
Keywords
- asymptotic density
- random variable
- statistical convergence
- statistical convergence in probability
- statistical convergence in mean of order r
- statistical convergence in distribution