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\ .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.

The sequence χ is statistically convergent to L if for each e > 0, lim η -1 {the number of k < τι :It is known that this method of summability cannot be included by any matrix method, but for bounded sequences it is included by the Cesáro matrix method C\. In this paper these results eure extended by comparing statistical convergence with the intersection of a collection Τ of summability matrices, each of which is somewhat like C\. It is shown that a bounded sequence is statistically convergent if and only if it is summable by every matrix in T. On the other hand, no countable collection of matrices can include statistical convergence for unbounded sequences. Also, the class Τ is studied to determine which classical summability matrices belong to T.

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\ .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.

Let S be the set of subsequences (xn k ) of a given real sequence (xn) which preserve the set of statistical cluster points. It has been recently shown that S is a set of full (Lebesgue) measure. Here, on the other hand, we prove that S is meager if and only if there exists an ordinary limit point of (xn) which is not also a statistical cluster point of (xn). This provides a non-analogue between measure and category.2010 Mathematics Subject Classification. Primary: 40A35. Secondary: 11B05, 54A20.

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