On Statistical Boundedness of Metric Valued Sequences

Annals of the University of Craiova Mathematics and Computer Science Series

In this paper using a non-negative regular summability matrix A and a non-trivial admissible ideal I in N we study some basic properties of strong AI-statistical convergence and strong AI-statistical Cauchyness of sequences in probabilistic metric spaces not done earlier. We also introduce the notion of strong AI∗-statistical Cauchyness and study its relationship with strong AI-statistical Cauchyness. Further we study some basic properties of strong AI-statistical limit points and strong AI-statistical cluster points of a sequence in probabilistic metric spaces.

2016

In this paper we define concepts of statistically convergent and statistically Cauchy multiple sequences in probabilistic normed spaces. We prove a useful characterization for statistically convergent multiple sequences. We will introduce the statistical limit points, statistical cluster points in probabilistic normed spaces. Moreover we will give the relation between them and limit points of multiple sequences in probabilistic normed spaces.

arXiv (Cornell University), 2022

In this paper using a non-negative regular summability matrix A and a non-trivial admissible ideal I in N we study some basic properties of strong A I -statistical convergence and strong A I -statistical Cauchyness of sequences in probabilistic metric spaces not done earlier. We also introduce strong A I * -statistical Cauchyness in probabilistic metric space and study its relationship with strong A I -statistical Cauchyness there. Further, we study some basic properties of strong A I -statistical limit points and strong A I -statistical cluster points of a sequence in probabilistic metric spaces.

2016

In this paper mainly, Wijsman deferred statistical convergence of sequence of sets in an arbitrary metric space is defined and some basic theorems are given. Besides new results, some results in this paper are the generalization of the results given in [3], [15] and [18].

Mathematical and Computer Modelling, 2011

A subset E of a metric space (X, d) is totally bounded if and only if any sequence of points in E has a Cauchy subsequence. We call a sequence (x n) statistically quasi-Cauchy if st − lim n→∞ d(x n+1 , x n) = 0, and lacunary statistically quasi-Cauchy if S θ − lim n→∞ d(x n+1 , x n) = 0. We prove that a subset E of a metric space is totally bounded if and only if any sequence of points in E has a subsequence which is any type of the following: statistically quasi-Cauchy, lacunary statistically quasi-Cauchy, quasi-Cauchy, and slowly oscillating. It turns out that a function defined on a connected subset E of a metric space is uniformly continuous if and only if it preserves either quasi-Cauchy sequences or slowly oscillating sequences of points in E.

In this paper we consider the notion of strongly I-statistically pre-Cauchy double sequences in probabilistic metric spaces in line of Das et. al. (On I-statistically pre-Cauchy sequences, Taiwanese J. Math 18 (1) (2014), 115-126) and introduce the new concept of strongly I *-statistically pre-Cauchy double sequences in probabilistic metric spaces. We mainly study interrelationship among strong I-statistical convergence, strong I-statistical pre-Cauchy condition and strong I *-statistical pre-Cauchy condition for double sequences in probabilistic metric spaces and examine some basic properties of these notions.

Proyecciones (Antofagasta), 2019

Boletim da Sociedade Paranaense de Matemática

In this paper we study some basic properties of strong λ-statistical convergence of sequences in probabilistic metric spaces. Also introducing the concept of strong λ-statistically Cauchy sequences we study its relationship with strong λ-statistical convergence in a probabilistic metric space. Further introducing the notions of strong λ-statistical limit point and strong λ-statistical cluster point of a sequence in a probabilistic metric space we examine their interrelationship.

2015

In this paper we construct some generalized new difference statistically convergentsequence spaces defined by a Musielak-Orlicz function over n − normed spaces. Wealso study several properties relevant to topological structures and inclusion relationsbetween these spaces.

Acta Mathematica Vietnamica, 2013

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Analysis, 2014

In this paper we study the concept of almost asymptotically lacunary statistical convergent sequences in probabilistic normed spaces and prove some basic properties.

2018

In this paper we consider the notion of strongly I-statistically pre-Cauchy double sequences in probabilistic metric spaces in line of Das et. al. (On I-statistically preCauchy sequences, Taiwanese J. Math 18 (1) (2014), 115–126) and introduce the new concept of strongly I∗-statistically pre-Cauchy double sequences in probabilistic metric spaces. We mainly study interrelationship among strong I-statistical convergence, strong I-statistical pre-Cauchy condition and strong I∗-statistical pre-Cauchy condition for double sequences in probabilistic metric spaces and examine some basic properties of these notions.

arXiv: Functional Analysis, 2016

In this paper we consider the notion of strong $I$-statistically pre-Cauchy double sequences in probabilistic metric spaces in line of Das et. al. [6] and introduce the new concept of strong $I^*$-statistically pre-Cauchy double sequences in real line as well as in probabilistic metric spaces. We mainly study inter relationship among strong $I$-statistical convergence, strong $I$-statistical pre-Cauchy condition and strong $I^*$-statistical pre-Cauchy condition for double sequences in probabilistic metric spaces and examine some basic properties of these notions.

2015

In this paper we study the concept of lacunary statistical convergent triple sequences in probabilistic normed spaces and prove some basic properties.

2014

In this paper we have introduced the concept of statistically convergent sequence in case of cone metric space and constructed statistically convergent, Cauchy and complete cone metric space and some theorems based on them. Consequently we have generalised several results in cone metric spaces from metric spaces.