A measure theoretical subsequence characterization of statistical convergence

Miller, Harry I.

doi:10.1090/S0002-9947-1995-1260176-6

A measure theoretical subsequence characterization of statistical convergence

Author: Harry I. Miller
Journal: Trans. Amer. Math. Soc. 347 (1995), 1811-1819
MSC: Primary 40C05; Secondary 40A99, 40D25
DOI: https://doi.org/10.1090/S0002-9947-1995-1260176-6
MathSciNet review: 1260176
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Abstract: The concept of statistical convergence of a sequence was first introduced by H. Fast. Statistical convergence was generalized by R. C. Buck, and studied by other authors, using a regular nonnegative summability matrix in place of ${C_1}$ .

The main result in this paper is a theorem that gives meaning to the statement: $S = \{ {s_n}\}$ converges to statistically if and only if "most" of the subsequences of converge, in the ordinary sense, to . Here is a regular, nonnegative and triangular matrix.

Corresponding results for lacunary statistical convergence, recently defined and studied by J. A. Fridy and C. Orhan, are also presented.

[1] Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
[2] Patrick Billingsley, Probability and measure, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Wiley Series in Probability and Mathematical Statistics. MR 534323
[3] R. Creighton Buck, Generalized asymptotic density, Amer. J. Math. 75 (1953), 335–346. MR 0054000, https://doi.org/10.2307/2372456
[4] Yuan Shih Chow and Henry Teicher, Probability theory, Springer-Verlag, New York-Heidelberg, 1978. Independence, interchangeability, martingales. MR 513230
[5] J. S. Connor, The statistical and strong 𝑝-Cesàro convergence of sequences, Analysis 8 (1988), no. 1-2, 47–63. MR 954458
[6] Jeff Connor, On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull. 32 (1989), no. 2, 194–198. MR 1006746, https://doi.org/10.4153/CMB-1989-029-3
[7] Jeff Connor, Two valued measures and summability, Analysis 10 (1990), no. 4, 373–385. MR 1085803, https://doi.org/10.1524/anly.1990.10.4.373
[8] Richard G. Cooke, Infinite matrices and sequence spaces, Dover Publications, Inc., New York, 1965. MR 0193472
[9] H. Fast, Sur la convergence statistique, Colloquium Math. 2 (1951), 241–244 (1952) (French). MR 0048548
[10] J. J. Sember and A. R. Freedman, On summing sequences of 0’s and 1’s, Rocky Mountain J. Math. 11 (1981), no. 3, 419–425. MR 722575, https://doi.org/10.1216/RMJ-1981-11-3-419
[11] A. R. Freedman and J. J. Sember, Densities and summability, Pacific J. Math. 95 (1981), no. 2, 293–305. MR 632187
[12] J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–313. MR 816582, https://doi.org/10.1524/anly.1985.5.4.301
[13] J. A. Fridy and H. I. Miller, A matrix characterization of statistical convergence, Analysis 11 (1991), no. 1, 59–66. MR 1113068, https://doi.org/10.1524/anly.1991.11.1.59
[14] J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993), no. 1, 43–51. MR 1227502
[15] -, Lacunary statistical summability (to appear).
[16] I. J. Schoenberg, The integrability of certain functions and related summability methods., Amer. Math. Monthly 66 (1959), 361–375. MR 0104946, https://doi.org/10.2307/2308747