Statistical Convergence and Cauchy Sequences in Quasi-Metric Spaces

Abstract

In this paper, we investigate statistical convergence and statistical Cauchy sequences in quasi-metric spaces by means of the asymptotic density of subsets of natural numbers. Our primary objective is to introduce a statistical analogue of the Cauchy definition given by Doitchinov (Topol Appl 30:127–148, 1988) and to examine its relationship with statistical convergence in asymmetric settings. We establish a necessary and sufficient condition for a sequence to be statistically Cauchy in terms of its dense subsequences, thereby extending classical subsequence characterizations to the quasi-metric framework. It is shown that, unlike several existing notions of Cauchy sequences in quasi-metric spaces, the proposed definition coincides with the standard statistical Cauchy criterion in the metric case.