Sanjoy Ghosal

On generalizations of certain summability methods using ideals

In this paper, following the line of Savas and Das (2011) [10], we provide a new approach to two well-known summability methods by using ideals, introduce new notions, namely, I-statistical convergence and I-lacunary statistical convergence, investigate their relationship, and make some observations about these classes.

Some further results on I-Cauchy sequences and condition (AP)

In this paper we provide answers to two important questions regarding I and I^*-Cauchy sequences introduced and studied by Nabiev et al. (2007) [9] which were left unanswered. We then introduce the ideas of I and I^*-divergent sequences in a metric space and study their certain properties. Our investigation strengthens and reconfirms importance of condition (AP) in the study of summability through ideals.

Generalized weighted random convergence in probability

The definition of weighted statistical convergence was first introduced by Karakaya & Chishti (2009) and later on Mursaleen et al. (2012) modified the definition of this concept. Using the modified definition, Edely et al. (2013) had researched further in approximation theory for periodic functions and Belen & Mohiuddine (2013) had further extended to the new-definition of weighted λ-statistical convergence. But some problems are still there, so the definition of weighted λ-statistical convergence is needed to modify. In this paper, we will introduce some new constraints which will make the definition of weighted λ-statistical convergence is more useful by using the definition of αβ-statistical convergence. Using it and independently, some newly developed concepts of the convergence of a sequence of random variables in probability, namely, weighted modulus αβ-statistical convergence of order γ, weighted modulus αβ-strong Cesaro convergence of order γ, weighted modulus S α β -convergence of order γ, and weighted modulus N α β -convergence of order γ, have been introduced and their basic interrelations also have been investigated.

Statistical convergence of order α in probability

Abstract In this paper the ideas of different types of convergence of a sequence of random variables in probability, namely, statistical convergence of order α in probability, strong p -Ces a ro summability of order α in probability, lacunary statistical convergence or S θ -convergence of order α in probability, and N θ -convergence of order α in probability have been introduced and their certain basic properties have been studied.

On Some Further Generalizations of Strong Convergence in Probabilistic Metric Spaces Using Ideals

Following the line of (Das et al., 2011, Savas and Das, 2011), we make a new approach in this paper to extend the notion of strong convergence and more general strong statistical convergence (Şencimen and Pehlivan, 2008) using ideals and introduce the notion of strong - and -statistical convergence and two related concepts, namely, strong -lacunary statistical convergence and strong --statistical convergence in a probabilistic metric space endowed with strong topology. We mainly investigate their interrelationship and study some of their important properties.

Extending asymmetric convergence and Cauchy condition using ideals

In this paper we use the notion of ideals to extend the convergence and Cauchy conditions in asymmetric metric spaces. The asymmetry (or rather, absence of symmetry) of these spaces makes the whole treatment different from the metric case and we use a genuinely asymmetric condition called (AMA) to prove many results and show that certain classic results fail in the asymmetric context if the assumption is dropped.

On asymmetric I and I∗-divergence

Abstract In this paper we introduce and study the concepts of I-divergence and I∗-divergence of sequences as well as double sequences in an asymmetric metric spaces. We investigate the interrelationship between I-divergence and I∗-divergence and show that they are equivalent under some condition and prove some basic properties of these concepts.

Characterization of rough weighted statistical limit set

Abstract Our focus is to generalize the definition of the weighted statistical convergence in a wider range of the weighted sequence {tn}n∈ℕ. We extend the concept of weighted statistical convergence and rough statistical convergence to renovate a new concept namely, rough weighted statistical convergence. On a continuation we also define rough weighted statistical limit set. In the year (2008) Aytar established the following results: The diameter of rough statistical limit set of a real sequence is ≤ 2r (where r is the degree of roughness) and in general it has no smaller bound. If the rough statistical limit set is non-empty then the sequence is statistically bounded. If x∗ and c belong to rough statistical limit set and statistical cluster point set respectively, then |x∗ − c| ≤ r. We investigate whether the above mentioned three results are satisfied for rough weighted statistical limit set or not? Answer is no. So our main objective is to interpret above mentioned different behaviors of the new convergence and characterize the rough weighted statistical limit set. Also we show that this set satisfies some topological properties like boundedness, compactness, path connectedness etc.

I-uniform continuity and I-uniform boundedness of a function

Abstract The concepts of I-convergence and I-Cauchy condition are a generalization of statistical convergence and statistical Cauchy conditions and are dependent on the notion of the ideal I of subsets of the set ℕ of positive integers. In this paper, we shall introduce two new notions of I-uniform continuity and I-uniform boundedness of a function with values in ℝ or in a metric space and then study their basic properties.