Lacunary ideal convergence in probabilistic normed spaces

TL;DR

This paper introduces and studies lacunary ideal convergence in probabilistic normed spaces, defining related limit and cluster points, and establishing their relationships, demonstrating a more general convergence method.

Contribution

It extends lacunary ideal convergence to probabilistic normed spaces, defining new concepts like lacunary $I$-limit points and lacunary $I$-Cauchy sequences, and shows its generality.

Findings

Lacunary $I$-convergence is established in probabilistic normed spaces.

Relations between lacunary $I$-limit points and cluster points are proved.

An example demonstrates the broader applicability of the proposed convergence method.

Abstract

An ideal $I$ is a family of subsets of positive integers $\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_k)$ of real numbers is said to be lacunary $I$-convergent to a real number $\ell$, if for each $ \varepsilon> 0$ the set $$\left\{r\in \mathbb{N}:\frac{1}{h_r}\sum_{k\in J_r} |x_{k}-\ell|\geq \varepsilon\right\}$$ belongs to $I.$ The aim of this paper is to study the notion of lacunary $I$-convergence in probabilistic normed spaces as a variant of the notion of ideal convergence. Also lacunary $I$-limit points and lacunary $I$-cluster points have been defined and the relation between them has been established. Furthermore, lacunary-Cauchy and lacunary $I$-Cauchy sequences are introduced and studied. Finally, we provided example which shows that our method of convergence in probabilistic normed spaces is more general.