Wijsman statistical and strong Cesàro convergence of closed sets via power-type Musielak–Orlicz Modulars

Abstract

We study convergence of sequences of nonempty closed sets in a metric space through the power-type Musielak–Orlicz modular Φn (t) = tp n . We introduce Wijsman (pn )-statistical convergence, Wijsman (pn)-strong Cesàro convergence, and a lacunary block-mean statistical version associated with a lacunary sequence θ = (kr). In the bounded-exponent case 1 ≤ pn ≤ p+ < ∞, we prove that, for Wijsman bounded sequences, these three modes of convergence are equivalent whenever the lacunary moderate growth condition holds. In the power-type setting this condition is equivalent to hr/kr → 0. We also give comparison results and examples showing why boundedness of the exponent sequence is essential. For random closed sets, we prove a pathwise Cesàro transfer result: at a fixed base point, almost sure modular strong Cesàro control, together with almost sure boundedness of the distance evaluations and bounded exponents, yields almost sure convergence of the Cesàro averages of the distance functionals to the Wijsman limit.