The sequence X is statisticallv convergent to L if for eacli e > 0, lim" number o{ k < ti : | -L | > e} = 0; a: is a statisticallv Cauchv sequence if for each € > 0 there is a positive integer N = N{e) such thatlim" -{the number of k < n : | -| > t} = 0. n These concepts are shown to be equivaleot. Also, Statistical convergence is studied as a regulär summability method, and it is shown that it cannot be included by any matrix metbod. There are two Tauberian theorems proved: one uses the Tauberian condition Axi, = 0(1/Ar), which is shown to be "best possible," and the other is concerned with gap sequences.AMS 1980 Classification numbers: 40A05, 40C05, 40D25, 40EX)5 FridyRecall (see, e.g., [2, p. 290]) that the "natural density" of a set K of positive integers is defined bywhere | < n : keK} \ denotes the number of elements of K not exceeding n. We shall be particularly concerned with integer sets having natural density zero. To facilitate this we introduce the following notation: if x is a sequence such that Xj^ satisfies property P for all k except a set of natural density zero, then we say that a:^ satisfies P for "almost all k ", and we abbreviate this by "a.a. k."

The sequence x is statistically convergent to L provided that for each ε > 0, lim «~" 1 {the number of k < n: \x^ -L\ > ε} = 0.n In this paper we study a related concept of convergence in which the set {k: k < n) is replaced by {k: k r -\ < k < k r }, for some lacunary sequence {k r } . The resulting summability method is compared to statistical convergence and other summability methods, and questions of uniqueness of the limit value are considered.

Abstract. Following the concept of statistical convergence and statistical cluster points of a sequence x, we give a definition of statistical limit superior and inferior which yields natural relationships among these ideas: e.g., x is statistically convergent if and only if st-liminfx = st-limsupx. The statistical core of x is also introduced, for which an analogue of Knopp's Core Theorem is proved. Also, it is proved that a bounded sequence that is C 1 -summable to its statistical limit superior is statistically convergent.

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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.