DOI: 10.1515/ms-2017-0152 Math. Slovaca 68 (2018), No. 4, 881–896 CHARACTERIZATION OF ROUGH WEIGHTED STATISTICAL LIMIT SET Pratulananda Das* — Sanjoy Ghosal** — Avishek Ghosh* — Sumit Som* (Communicated by J´an Bors´ ık ) ABSTRACT. Our focus is to generalize the definition of the weighted statistical convergence in a wider range of the weighted sequence {tn} n∈N . We extend the concept of weighted statistical conver- gence and rough statistical convergence to renovate a new concept namely, rough weighted statistical convergence. On a continuation we also define rough weighted statistical limit set. In the year (2008) Aytar established the following results: (i) The diameter of rough statistical limit set of a real sequence is ≤ 2r (where r is the degree of roughness) and in general it has no smaller bound. (ii) If the rough statistical limit set is non-empty then the sequence is statistically bounded. (iii) If x∗ and c belong to rough statistical limit set and statistical cluster point set respectively, then |x∗ - c|≤ r. We investigate whether the above mentioned three results are satisfied for rough weighted statistical limit set or not? Answer is no. So our main objective is to interpret above mentioned different behaviors of the new convergence and characterize the rough weighted statistical limit set. Also we show that this set satisfies some topological properties like boundedness, compactness, path connectedness etc. c 2018 Mathematical Institute Slovak Academy of Sciences 1. Introduction Since 1951 when Fast [9] and Steinhaus [30] defined statistical convergence for real sequences (though the concept was known to Zygmund way back in 1935), several generalizations and ap- plications of this notion have been investigated (see [5]–[7],[10]–[12],[17]–[21],[28]–[27] where many more references can be found). The idea of rough convergence is a generalization of the ordinary convergence, which was first defined in 2001 by Phu [23] in finite dimensional normed linear spaces. In his papers [24] and [25] related to the subject, he defined the rough continuity of linear operators and rough convergence in infinite dimensional spaces respectively. In particular an interesting generalization of rough convergence was introduced by Aytar [2], by using the notion of natural density of the set N of positive integers which he named as rough statistical convergence (more results on this convergence can be found in [1,26]). The theory of rough convergence has been discussed in Fuzzy set theory [3], Probability theory [8], and so on. 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 40A05, 40A35; Secondary 40G15. K e y w o r d s: rough statistical convergence, weighted statistical convergence, rough weighted statistical convergence, rough weighted statistical limit set. 881 Brought to you by | University of Sussex Library Authenticated Download Date | 8/12/18 4:27 PM