Lacunary statistical convergence in topological groups. (English) Zbl 0835.43006

A sequence \((x(k))\) in a topological group \(X\) is called statistically convergent to an element \(l\) of \(X\) if for each neighbourhood \(U\) of 0 \(\lim_{m \to \infty} m^{-1} |\{k \leq m : x(k) - l \notin U\} |= 0\) where the vertical bars indicate the number of elements in the enclosed set. This concept was first given by H. Fast [Colloq. Math. 2, 141-144 (1951; Zbl 0044.336)] and studied by J. A. Fridy [Analysis 5, 301-313 (1985; Zbl 0588.40001)], K. Kolk [Tartu Ul. Toimetised 928, 41-52 (1991)] and I. J. Maddox [Math. Proc. Camb. Philos. Soc. 83, 61-64 (1978; Zbl 0392.40001); 104, 141-145 (1988; Zbl 0674.40008)]. J. A. Fridy and C. Orhan [Pac. J. Math. 160, 43- 51 (1993; Zbl 0794.60012)] studied lacunary statistical convergence by giving the definition as follows: a sequence \((x(k))\) of real or complex numbers is said to be lacunary statistically convergent to a number \(L\) if for each \(\varepsilon > 0\) \[ \lim_{r \to \infty} (h_r)^{-1} \Bigl |\biggl \{k \in I_r : \bigl |x(k) - L \bigr |\geq \varepsilon \biggr\} \Bigr |= 0. \] The purpose of this paper is to introduce lacunary statistical convergence in metrizable topological groups and to prove some inclusion theorems between the set of all statistical convergent sequences and the set of all lacunary statistically convergent sequences.


43A55 Summability methods on groups, semigroups, etc.