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.

We study the concepts of I-limit and I-cluster points of a sequence, where I is an ideal with the Baire property. We obtain the relationship between I-limit and I-cluster points of a subsequence of a given sequence and the set of its classical limit points in the sense of category theory.

Abstract. In this paper, we present some relationships between convergence and uniform statistical convergence of a given sequence and its subsequences. The results concerning uniform statistical convergence presented here are also closely related to earlier results regarding statistical convergence and almost convergence of sequences, and are dealing with measure and in a minor case with category. Finally, we present a Cauchy type characterization of uniform statistical convergence and a result concerning uniform statistical convergence of subseries of a series.

J.A. Fridy [Statistical limit points, Proc. Amer. Math. Soc., 1993] considered statistical cluster points and statistical limit points of a given sequence x. Here we show that almost all subsequences of x have the same statistical cluster point set as x. Also, we show an analogous result for the statistical limit points of x.

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