Lacunary Statistical Limit Points in Random 2-Normed Spaces

We introduce the notion -cluster points, investigate the relation between -cluster points and limit points of sequences in the topology induced by random 2-normed spaces, and prove some important results. 1. Introduction and Background An interesting and important generalization of the notion of metric space was introduced by Menger [1] under the name of statistical metric space, which is now called probabilistic metric space. In this theory, the notion of distance has a probabilistic nature. Namely, the distance between two points and is represented by a distribution function ; and for , the value is interpreted as the probability that the distance from to is less than . In fact the probabilistic theory has become an area of active research for the last forty years. An important family of probabilistic metric spaces are probabilistic normed spaces. The notion of probabilistic normed spaces was introduced in [2] and further it was extended to random/probabilistic 2-normed spaces by Gole？ [3] using the concept of 2-norm of G？hler [4]. Applications of this concept have been investigated by various authors, for example, [5–7]. The concept of statistical convergence for sequences of real number was introduced by Fast in [8] and Steinhaus in [9] independently in the same year 1951. A lot of developments have been made in this area after the works of Salat [10] and Fridy [11]. Recently, Mohiuddine and Aiyub [12] studied lacunary statistical convergence as generalization of the statistical convergence and introduced the concept -statistical convergence in random 2-normed space. In [13], Mursaleen and Mohiuddine extended the idea of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. Also lacunary statistically convergent double sequences in probabilistic normed space was studied by Mohiuddine and Sava？ in [14]. The aim of this work is to introduce and investigate the relation between -statistical cluster points, -statistical limit points, and ordinary limit points of sequence in random 2-normed spaces. First, we recall some of the basic concepts that will be used in this paper. All the concepts listed below are studied in depth in the fundamental book by Schweizer and Sklar [2]. Let denote the set of real numbers and . A mapping is called a distribution function if it is nondecreasing and left continuous with and . We denote the set of all distribution functions by such that . If , then , where It is obvious that for all . A triangular norm ( -norm) is a continuous mapping such that is an abelian monoid with unit one and if