Lacunary ideal convergence of multiple sequences in probabilistic normed spaces

Abstract

An ideal I is a family of subsets of positive integers N×N which is closed under finite unions and subsets of its elements. The aim of this paper is to study the notion of lacunary I-convergence of double sequences in probabilistic normed spaces as a variant of the notion of ideal convergence. Also lacunary I-limit points and lacunary I-cluster points have been defined and the relation between them has been established. Furthermore, lacunary-Cauchy and lacunary I-Cauchy, lacunary I*-Cauchy, lacunary I*-convergent double sequences are introduced and studied in probabilistic normed spaces. Finally, we provided example which shows that our method of convergence in probabilistic normed space is more general.