DOI: 10.1515/ms-2017-0275 Math. Slovaca 69 (2019), No. 4, 871–890 WHEN DEVIATION HAPPENS BETWEEN ROUGH STATISTICAL CONVERGENCE AND ROUGH WEIGHTED STATISTICAL CONVERGENCE Sanjoy Ghosal* — Avishek Ghosh** (Communicated by J´an Bors´ ık ) ABSTRACT. In this paper we introduce rough weighted statistical limit set and weighted statistical cluster points set which are natural generalizations of rough statistical limit set and statistical cluster points set of double sequences respectively. Some new examples are constructed to ensure the deviation of basic results. Both the sets don’t follow the usual extension properties which will be discussed here. c 2019 Mathematical Institute Slovak Academy of Sciences 1. Introduction The idea of statistical convergence for double sequences was introduced by Mursaleen and Edely [23] as follows: Let K ⊂ N × N be a two-dimensional set of positive integers and let K(m, n) be the numbers of (i, j ) ∈ K such that i ≤ m and j ≤ n. Then the two-dimensional analogue of natural density can be defined by d 2 (K)= lim m,n→∞ K(m, n) mn . A double sequence x = {x ij } i,j∈N in a norm linear space (X, ‖.‖) is said to be statistically convergent to c ∈ X if for every ε> 0, d 2 ({(i, j ) ∈ N × N : ‖x ij − c‖≥ ε})=0 and we write x ij st2 −−→ c. Several works on this convergence are done later on which can be seen from [1, 4, 12, 32]. A double sequence x = {x ij } i,j∈N in X is said to be statistically bounded [13, 22] if there exists a positive real number G such that d 2 ({(i, j ) ∈ N × N : ‖x ij ‖≥ G})=0. An element y ∈ X is called a statistical cluster point [11,22] of a double sequence x = {x ij } i,j∈N in X provided d 2 ({(i, j ) ∈ N × N : ‖x ij − y‖ <ε}) =0, 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 40A35; Secondary 40G15. K e y w o r d s: double weighted natural density, rough weighted statistical convergence, rough weighted statistical limit set, weighted statistical cluster points. The research of the second author is supported by Jadavpur University, Kolkata-700032, West Bengal, India. 871 Brought to you by | Uppsala University Library Authenticated Download Date | 12/7/19 10:37 PM