Characterization of statistical convergence of complex uncertain double sequence
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Characterization of statistical convergence of complex uncertain double sequence Birojit Das1 · Binod Chandra Tripathy2 · Piyali Debnath1 · Baby Bhattacharya1 Received: 1 September 2020 / Revised: 18 October 2020 / Accepted: 21 October 2020 © Springer Nature Switzerland AG 2020
Abstract The main aim of this paper is to introduce statistical convergence of a complex uncertain double sequence. Characterization of statistical convergence is given via uncertain measure, uncertain expected value operator, uncertain distribution function and also with respect to almost surely, uniformly almost surely. The concept of boundedness is introduced to establish some results. Finally, the notion of statistically complex uncertain double Cauchy sequence is proposed to obtain interrelationship with statistically convergent complex uncertain double sequence. Keywords Uncertainty space · Complex uncertain double sequence · Statistical convergence · Complex uncertain Cauchy double sequence · Bounded complex uncertain double sequence Mathematics Subject Classification 60B10 · 60E05 · 40A05 · 40A30 · 40F05
1 Introduction Convergence of sequences [17,21] plays a pivotal role in the study of fundamental theory of functional analysis and in particular sequence space. The concept of statistical convergence of real and complex sequences was introduced in the mid of twentieth
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Piyali Debnath [email protected] Birojit Das [email protected] Binod Chandra Tripathy [email protected] Baby Bhattacharya [email protected]
1
Department of Mathematics, NIT Agartala, Agartala, Tripura, India
2
Department of Mathematics, Tripura University, Agartala, Tripura, India 0123456789().: V,-vol
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century by Fast [6], Buck [2], Schohenberg [20] independently within a short period of time. But the research on this concept got flourish soon after the works of Salat [18] and Fridy [7] came into literature. The study of statistical convergence in double sequence has been initiated by Tripathy [23], Mursaleen and Edely [12], Moricz [11] separately at the same time. To start with statistical convergence, it is needed to give a brief idea about the concept of natural density [2] on a subset of natural numbers. Let A be any subset of the set of natural numbers N. Then the natural density of A is denoted by δ(A) and is defined by δ(A) = lim
n→∞
1 |{k ≤ n : k ∈ A}|, n
where the vertical bars stands for cardinality of the set A. For example, the set 2Z of even integers has natural density 21 and the set of primes has natural density zero. The notion of natural density of a subset K of N × N is defined as follows: δ2 (K ) =
lim
n,m→∞
K (n, m) (Limit taken in Pringsheim’s sense) nm
and statistical convergence of a double sequence is as follows: A double sequence x = {x jk } of real or complex terms is said to be statistically convergent to the number l if for each ε > 0, the set {( j, k) : j ≤ n, k ≤ m : |x jk − l| ≥ ε} has double natural density zero. The notion of generalized statistical convergence of dou
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