Some convergence types in probabilistic g-metrik spaces

In the exploration of probabilistic G-metric spaces, we introduce a novel and intriguing concept known as lacunary statistical convergence specifically tailored for double sequences. This concept serves as a powerful analytical tool, enabling us to delve into the intricate dynamics of double sequences within the probabilistic G-metric framework. Our study not only introduces this concept but also rigorously establishes the foundational properties inherent in double lacunary convergent sequences. These properties form the backbone of our exploration, providing a solid theoretical underpinning for the subsequent analyses. Furthermore, we go beyond mere theoretical development by presenting a necessary condition for lacunary convergence of double sequences in probabilistic generalized metric spaces. It is essential to note that this condition, although necessary, may not be sufficient, emphasizing the delicate balance and nuanced nature of probabilistic G-metric spaces. To illustrate the practical applicability and significance of our proposed concept and conditions, we offer a rich collection of examples. These real-world instances showcase the versatility and utility of our approach in addressing a diverse array of scenarios within probabilistic G-metric spaces. Taking our exploration a step further, we incorporate the concept of an ideal, introducing new convergence types for double sequences in probabilistic G-metric spaces. These novel convergence types broaden the scope of our study, revealing additional layers of complexity and richness within the probabilistic G-metric framework. The relationships among the newly introduced convergence types are carefully examined and expressed through theorems, providing a comprehensive understanding of their interplay and dependencies. This rigorous theoretical framework adds depth and clarity to our exploration, facilitating a more profound comprehension of the intricacies involved. In the realm of mathematical proofs, our methodology is multifaceted. We employ both direct proof methods and proof by contradiction, leveraging these approaches strategically to establish the validity of our propositions and theorems. This dual strategy enhances the robustness and reliability of our mathematical analyses, ensuring the integrity of our contributions to the field of probabilistic G-metric spaces.