A Measure Theoretical Subsequence Characterization of Statistical Convergence

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\ . The main result in this paper is a theorem that gives meaning to the statement: S= {sn} converges to L statistically (T) if and only if "most" of the subsequences of 5 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. Introduction The concept of the statistical convergence of a sequence of reals S = {s„} was first introduced by H. Fast [9]. The sequence S = {s„} is said to converge statistically to L and we write lim sn = L (stat) if for every e > 0, n—»oo lim n~x\{k < n : \sk L\ > e}\ = 0, n—>oo where \A\ denotes the cardinality of the set A. Properties of statistically convergent sequences were studied in [5, 6, 12, and 16]. In [13] Fridy and Miller gave a characterization of statistical convergence for bounded sequences using a family of matrix summability methods. Statistical convergence can be generalized by using a regular nonnegative summability matrix A in place of C\. This idea was first mentioned by R. C. Buck [3] in 1953 and has been further studied by Sember and Freedman ([10 and 11]) and Connor ([5 and 7]). Regular nonnegative summability matrices turn out to be too general for our purposes here, instead we use the concept of a mean. A matrix T = (amn ) will be called a mean if amn > 0 when n m, Y^=\ amn = 1 for all m and limm_0O amn = 0 for each n . If T = (amn) is a mean, following Buck, a sequence S = {sn} is said to be statistically T-summable to L and we write sn -> L (stat T) if for every e > 0 Received by the editors August 18, 1993 and, in revised form, February 14, 1994; originally communicated to the Proceedings oftheAMS by Andrew Bruckner. 1991 Mathematics Subject Classification. Primary 40D25; Secondary 40G99, 28A12. ©1995 American Mathematical Society 0002-9947/95 $1.00+ $.25 per page