Statistical convergence of order α\alpha in probability

In this paper ideas of different types of convergence of a sequence of random variables in probability, namely, statistical convergence of order $\alpha$ in probability, strong $p$-Ces\grave{\mbox{a}}ro summability of order $\alpha$ in probability, lacunary statistical convergence or $S_{\theta}$-convergence of order $\alpha$ in probability, ${N_{\theta}}$-convergence of order $\alpha$ in probability have been introduced and their certain basic properties have been studied.Comment: Vol- 21, Issue-2, pages 253-265 in Arab Journal of Mathematical Sciences (2015

When I-Cauchy nets in complete uniform spaces are I-convergent

AbstractIn this paper we continue our investigation of nets using ideals in line of our earlier work where we had studied I-Cauchy nets and asked when I-Cauchy nets in complete uniform spaces are I-convergent in line of a problem by Di Maio and Kočinac who asked in 2008 when statistically Cauchy sequences are statistically convergent in uniform spaces. To answer this, here we consider another type of Cauchy condition of nets, namely I⁎-Cauchy condition and examine its basic properties and in particular its relation with the concept of I-Cauchy nets. This helps us to give an answer to the above mentioned open question

On generalizations of certain summability methods using ideals

AbstractIn 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

Sλ-convergence of a sequence of random variables

AbstractIn this paper we continue our investigation of recent notions of λ-statistical convergence in probability and λ-statistical convergence in mean of order r in Ghosal (2014) [1] and introduce the notion of λ-statistical convergence in distribution. We mainly investigate their interrelationship and study some of their important basic properties

Weighted statistical convergence of order α and its applications

The definition of weighted statistical convergence was first introduced by Karakaya and Chishti (2009) [1]. After that the definition was modified by Mursaleen et al. (2012) [2]. But some problems are still there; so it will be further modified in this paper. Using it some newly developed definitions of the convergence of a sequence of random variables in probability have been introduced and their interrelations also have been investigated, and in this way a partial answer to an open problem posed by Das and Savas (2014) [3] has been given

Weighted modulus Sθ-convergence of order α in probability

Let θ={kr}r∈N∪{0} be a lacunary sequence, ϕ be a modulus function and {tn}n∈N be a sequence of real numbers such that tn>δ,∀n∈N (where δ is a fixed positive real number) and Tn=t1+t2+⋯+tn (where n∈N and T0=0). A sequence of random variables {Xn}n∈N is said to be weighted modulus Sθ-convergent of order α in probability (where 00, limr→∞1(Tkr−Tkr−1)α|{k∈(Tkr−1,Tkr]:tkϕ(P(|Xk−X|≥ε))≥δ}|=0. The results are applied to build the probability distribution for weighted modulus Nθ-convergence of order α. Also these methods are compared with the convergence of weighted modulus statistical convergence of order α and weighted modulus strong Cesàro convergence of order α respectively. If limsupr→∞TkrTkr−1α<∞, then weighted modulus Sθ-convergence of order α in probability implies weighted modulus statistical convergence of order α in probability and weighted modulus Nθ-convergence of order α implies weighted modulus strong Cesàro convergence of order α in probability except the condition limsupr→∞TkrTkr−1α=∞. So our main objective is to interpret the above exceptional condition and produce a relational behavior of above mentioned four convergences. This is also used to prove the uniqueness of limit value of weighted lacunary statistical convergence and improve the definition of weighted lacunary statistical convergence

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 (Şençimen and Pehlivan, 2008) using ideals and introduce the notion of strong ℐ- and ℐ*-statistical convergence and two related concepts, namely, strong ℐ-lacunary statistical convergence and strong ℐ-λ-statistical convergence in a probabilistic metric space endowed with strong topology. We mainly investigate their interrelationship and study some of their important properties

Influence of theta-Metric Spaces on the Diameter of Rough Weighted I-2-Limit Set

In this paper we continue our investigation of the recent summability notion introduced in [Math. Slovaca 69 (4) (2019) 871-890] (where rough weighted statistical convergence for double sequences is discussed over norm linear spaces) and introduce the notion of rough weighted I-2-convergence over theta-metric spaces. Also we exercise the behavior of weighted I-2-cluster points set over theta-metric spaces. Based on the new notion we vividly discuss some important results and perceive how the existing results are vacillating