ON STATISTICAL LIMIT POINTS

The set of all statistical limit points of a given sequence xn is characterized as an Fσ-set. It is also characterized in terms of discontinuity points of distribution functions of xn. Introduction Following the concept of a statistically convergent sequence, J. A. Fridy [Fr93] introduced the notion of a statistical limit point of a given sequence xn, n = 1, 2, . . . , of real numbers: A real number x is said to be a statistical limit point of the sequence xn if there exists a subsequence xkn , n = 1, 2, . . . , such that limn→∞ xkn = x and the set of indices kn has a positive upper asymptotic density (see Definitions and Notations). Fridy studied the set Λ(xn) of all such points. He has shown that this set need not be closed or open in R. In section 1 of the paper we prove that the set Λ(xn) is an Fσ-set in R for an arbitrary sequence xn and vice-versa for any given Fσ-set X there exists a sequence xn such that X = Λ(xn). In section 2 we prove that the set Λ(xn) coincides with the set of all discontinuity points of distribution functions of xn. This leads to some sufficient conditions for Λ(xn) = ∅. Finally, in section 3 we compute Λ(xn) using an explicit form of the set G(xn) of all distribution functions of xn for some special xn. Definitions and notations IfA ⊂ N, then we denote by |A| the cardinality ofA and A(n) = |{k ≤ n; k ∈ A}|. The numbers d(A) = lim inf n→∞ A(n) n , d(A) = lim sup n→∞ A(n) n are called the lower and upper asymptotic density of A, respectively. If there exists the limit limn→∞A(n)/n, then d(A) = d(A) = d(A) is said to be the asymptotic density of A (cf. [HR66, xix]). Received by the editors December 12, 1998 and, in revised form, January 10, 2000. 2000 Mathematics Subject Classification. Primary 40A05, 11K31, 11B05.