ON ℐ2 AND ℐ˚2-CONVERGENCE IN ALMOST SURELY OF COMPLEX UNCERTAIN DOUBLE SEQUENCES

Sep 10, 2026

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This study investigates ℐ2-convergence almost surely (a.s.) and ℐ˚2-convergence a.s. for complex uncertain double sequences in an uncertainty space. It analyzes key properties of these convergence notions and clarifies their interrelationships. The work further introduces ℐ2 and ℐ˚2-Cauchy sequences a.s. for complex uncertain double sequences and studies how these Cauchy concepts relate to the corresponding convergence definitions.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages17

Reading Time43 min

AccessFree to download

1. Introduction and Background

2. Preliminaries

DOI: [10.15393/j3.art.2023.12630](10.15393/j3.art.2023.12630) UDC 517.98, 515.124 ¨O . Ki¸si, M. Grdal ON ℐ2 AND ℐ˚2-CONVERGENCE IN ALMOST SURELY OF COMPLEX UNCERTAIN DOUBLE SEQUENCES Abstract. In this study, we investigate the notions of ℐ2-convergence almost surely (a.s.) and ℐ˚2-convergence a.s. of complex uncertain double sequences in an uncertainty space, and obtain some of their features and identify the relationships between them. In addition, we put forward the concepts of ℐ2 and ℐ˚2-Cauchy sequence a.s. of complex uncertain double sequences and investigate their relationships. Key words: uncertainty theory, complex uncertain variable,ℐ2-convergence, ℐ˚2-convergence 2020 Mathematical Subject Classification: 40A05, 40A35, 40G15, 60B10 1. Introduction and Background. The concept of statistical convergence was introduced by Fast [7] and then it was further investigated from the sequence-space point of view by several authors (see, for example, [8], [15]) . Statistical convergence has become one of the most active areas of research due to its wide applicability in various branches of mathematics, such as number theory, mathematical analysis, probability theory, etc. The study of statistical convergence of double sequence has been initiated by Moricz [16], Mursaleen and Edely [17], Tripathy [23], independently. The notion of the ideal convergence is a common generalization of the classical notions of convergence and statistical convergence. The notion of ℐ-convergent was further investigated from the sequencespace point of view and linked with the summability theory by Kostyrko et al. [12] . Later, this concept has been generalized in many directions. Das et al. [2] presented the notion of ℐ-convergence of double sequencesin a metric space and worked out some features of this convergence. More details on statistical convergence, ℐ-convergence, and on applications of © Petrozavodsk State University, 2023 this concept can be found in D¨undar and Altay [6], G¨urdal and Huban [9], G¨urdal and ¸Sahiner [10], Nabiev et al. [18], and Sava¸s and G¨urdal [21], [22] . Liu [13] was the first to introduce the uncertainty theory based on an uncertain measure that satisfies normality, duality, subadditivity, and product axioms. Nowadays uncertainty theory has become one of the most active areas of research due to its wide applicability in various domains such as uncertain programming, uncertain optimal control, uncertain risk analysis, uncertain differential equation, etc. For more details, one may refer to [14] . In order to identify complex uncertain sequences, the notion of uncertain variables was defined over the uncertain space. Complex uncertain sequences are measurable functions from an uncertain space to theset of all complex numbers C. Over the last few years, the study of convergence of sequences in the complex uncertain space has drawn attention of the researchers. In 2016, Chen [et. al](et. al). [1] investigated various types of convergence of sequences, such as convergence almost surely, convergence in measure, convergence in mean, and convergence in distribution, in the complex uncertain space. He mainly studied the interrelationship between the notions. The notion of statistical convergence was first developed in terms of complex uncertain sequences by Tripathy and Nath [24] . Later on, a lot of work has been carried out in this direction till today (see, for example,[3], [4], [5], [11], [19], [20], [24]) . The aim of this study is to present the notion of ℐ˚2-convergence almost surely in complex uncertain theory, examine several properties, and identify the relationships between ℐ2 and ℐ˚2-convergence almost surely of complex uncertain sequence. In addition, we put forward ℐ2 and ℐ˚2-Cauchy sequence almost surely of complex uncertain sequence. 2. Preliminaries In this section, we gather the necessary results and techniques on which we will rely to accomplish our main results. Utilizing the noti

FAQs

What convergence notions are studied for complex uncertain double sequences?

The document studies ℐ2-convergence almost surely (a.s.) and ℐ˚2-convergence a.s. in an uncertainty space, focusing on their properties and relationships.

How are ℐ2 and ℐ˚2-Cauchy sequences a.s. defined and used?

It proposes the concepts of ℐ2 and ℐ˚2-Cauchy sequence almost surely for complex uncertain double sequences, then investigates their connections with the corresponding convergence notions.

What background concepts from uncertainty theory and ideals are included?

The preliminaries gather results on ideals and filters (including strongly admissible ideals) and recall the definition of an uncertain measure via axioms, along with the setup of uncertainty spaces.

Sep 10, 2026

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