Window-weighted Wijsman convergence and pre-Cauchy subsequence criteria in idempotent bicomplex metric hyperspaces

Abstract

We study window-weighted Wijsman statistical convergence for sequences of non-empty closed sets in hyperspaces associated with idempotent bicomplex metric spaces. The ambient distance is assumed to have the form ? = d1 e1 + d2 e2, where d1 and d2 are ordinary metrics on the same underlying set. Thus the present paper concerns this idempotent, componentwise class of bicomplex-valued metrics, rather than arbitrary bicomplex-valued metric structures. For a closed set A, the point-to-set distance is represented by the two scalar profiles d1 (x, A) and d2 (x, A). We define window-weighted Wijsman statistical convergence and the corresponding pre-Cauchy condition by requiring simultaneous control of these two profiles along weighted moving windows. The main results give exact componentwise characterizations, establish invariance under weighted-equivalent changes of the window scheme, and show that a window-weighted Wijsman pre-Cauchy sequence becomes statistically convergent whenever it has an ordinarily Wijsman convergent trace of positive lower window weight. We also prove stability under negligible perturbations and show that ordinary Wijsman convergence on a trace of full window density determines the statistical convergence of the whole sequence. Examples illustrate the dependence on the window scheme, the strict weakness of the pre-Cauchy condition, and the effect of imposing simultaneous closedness with respect to two non-equivalent component metrics. © 2026 by the authors.