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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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 $ A$ 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 $ L$ statistically $ (T)$ if and only if "most" of the subsequences of $ S$ converge, in the ordinary sense, to $ L$. Here $ T$ 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.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1260176-6
Article copyright: © Copyright 1995 American Mathematical Society