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
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Trans. Amer. Math. Soc. 347 (1995), 1811-1819 Request permission

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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