Rough statistical convergence of a sequence of random variables in probability

Abstract

In this paper, following the works of Phu (Numer Funct Anal Optimiz 22:201–224, 2001), Aytar (Numer Funct Anal Optimiz 29(34):291–303, 2008) and Ghosal (Appl Math 58(4):423–437, 2013) we make a new approach to extend the application area of rough statistical convergence usually used in sequence of real numbers to the theory of probability distributions. The introduction of these concepts in probability of rough statistical convergence, rough strong Cesaro summable, rough lacunary statistical convergence, rough \(N_\theta \)-convergence, rough \(\lambda \)-statistical convergence and rough strong \((V,\lambda )\)-summable generalize the convergence analysis to accommodate any form of distribution of random variables. Among these six concepts in probability only three convergences are distinct- rough statistical convergence, rough lacunary statistical convergence and rough \(\lambda \)-statistical convergence where rough strong Cesaro summable is equivalent to rough statistical convergence, rough \(N_\theta \)-convergence is equivalent to rough lacunary statistical convergence, rough strong \((V,\lambda )\)-summable is equivalent to rough \(\lambda \)-statistical convergence. Basic properties and interrelations of above mentioned three distinct convergences are investigated and make some observations about these classes and in this way we show that rough statistical convergence in probability is the more generalized concept than usual rough statistical convergence.