On the role of the Ky Fan metric in rough ideal convergence in probability

TL;DR

This paper introduces rough ideal convergence in probability using the Ky Fan metric, unifying ideal convergence and convergence in probability, and explores properties of the limit sets and cluster points.

Contribution

It defines rough ideal convergence in probability with the Ky Fan metric and analyzes the topological structure of limit and cluster point sets.

Findings

Rough ideal limit set is closed and bounded in the Ky Fan metric.

Set of strong rough ideal cluster points is always closed.

Characterization of maximal admissible ideals via cluster points.

Abstract

Given a probability space $(S,\Delta, \mathbb{P})$ and a separable metric space $(U,d)$, the $Ky~Fan$ metric $\rho(X,Y)$ on the space $\mathfrak{X}^0$ of equivalence classes of random variables (w.r.t. almost sure equality) formed from the set $\mathfrak{X}(U)$ of $U$-valued random variables is given by $\rho(X,Y)=\inf \{\varepsilon>0:\mathbb{P}(d(X,Y)>\varepsilon)\leq\varepsilon\}.$ In this article, we primarily introduce the concept of rough ideal convergence in probability which serves as a unifying generalization of both ideal convergence of sequences in metric spaces and convergence of random variables in probability. We demonstrate that the rough ideal limit set is closed and bounded w.r.t. the $Ky~Fan$ metric $\rho$, and that, for a certain class of ideals, it forms an $F_{\sigma\delta}$ subset of $\mathfrak{X}^0$. In this process, we present the key concepts of strong and weak rough ideal cluster points in probability. It turns out that the set of strong rough ideal cluster points in probability is always closed, whereas the weak set is conditionally closed in the metric space ($\mathfrak{X}^0,\rho)$. Finally, we obtain a characterization of a maximal admissible ideal in terms of the sets of strong rough ideal cluster points and the rough ideal limit set in probability.