Statistical convergence of double sequences on probabilistic normed spaces defined by $[ V, \lambda, \mu ]$-summability

Pankaj Kumar,S S Bhatia,Vijay Kumar

doi:10.5269/bspm.v33i2.21670

Abstract

In this paper, we aim to generalize the notion of statistical convergence for double sequences on probabilistic normed spaces with the help of two nondecreasing sequences of positive real numbers $\lambda=(\lambda_{n})$ and $\mu = (\mu_{n})$ such that each tending to zero, also $\lambda_{n+1}\leq \lambda_{n}+1, \lambda_{1}=1,$ and $\mu_{n+1}\leq \mu_{n}+1, \mu_{1}=1.$ We also define generalized statistically Cauchy double sequences on PN space and establish the Cauchy convergence criteria in these spaces.

Highlights

Background and preliminariesFirst, We recall some notations and basic definitions those will be used in this paper
We normalize all distribution functions to be left continuous on unextended real line R = (−∞, +∞)
The maximal element for ∆+ in this order is the d.f. given by ε0(x) =

Summary

Introduction

Background and preliminariesFirst, We recall some notations and basic definitions those will be used in this paper. In 1993, using triangle functions, Alsina et al [1] defined probabilistic normed spaces as follows: Definition 2.2. The element p is called the statistical limit of the sequence (pn)n with respect to the probabilistic norm v and we write stv → lim pn = p

Objectives

Results

Conclusion