I-Statistical pre-Cauchy and frequent Cauchy criteria for compact sets in Hausdorff hyperspaces

Abstract

In this paper, we study pairwise asymptotic criteria for sequences of compact sets in the Hausdorff hyperspace of a metric space. We introduce the notions of I-statistically pre-Cauchy and frequent Cauchy compact-set sequences and examine their relation to convergence under suitable additional assumptions. After establishing basic permanence properties, we show that I-statistical convergence implies the corresponding pre-Cauchy condition. For bounded sequences, we obtain two explicit characterizations, one in terms of double means of Hausdorff distances and the other in terms of bounded moduli. We also investigate the relation between frequent Cauchy behavior and frequent convergence under compactness assumptions on the hyperspace or on the closure of the range of the sequence. Finally, we present criteria under which pairwise asymptotic conditions can be upgraded to actual I-statistical convergence, and we include examples that illustrate both the scope and the limitations of the pairwise approach in the hyperspace setting. © 2026 by the authors.