On the role of the Ky Fan metric in rough ideal convergence in probability
Tamim Aziz, Sanjoy Ghosal
TL;DR
The paper extends convergence concepts by introducing rough $\mathcal{I}^{\mathbb{P}}$-convergence in probability using the Ky Fan metric $\rho$, unifying ideal convergence with probabilistic convergence for sequences of $U$-valued random variables. It proves that $\rho$ metrizes convergence in probability and transfers rough convergence from deterministic sequences to associated random variables, with an equivalence between $\mathcal{I}^{\mathbb{P}}$-convergence in probability and $\mathcal{I}$-convergence in the $\rho$-norm. It then analyzes the limit sets $\mathcal{I}^{\mathbb{P}}{-LIM^rX_i}$, establishing a sharp diameter bound $diam_{\rho}\le\min\{1,2r\}$, closedness for analytic $P$-ideals, and a Borel $F_{\sigma\delta}$ structure, together with sandwich relations $\bar{\theta}_r(X_*)\subseteq \mathcal{I}^{\mathbb{P}}{-LIM^rX_i}\subseteq \bar{B}_r(X_*)$, including sharpness examples. The third part introduces rough $\mathcal{I}$-limit points and strong/weak rough cluster points in probability, proves nonemptiness and closedness under suitable conditions, and provides a maximal admissible ideal characterization via equality of $\mathcal{I}^{\mathbb{P}}{-LIM^rX_i}$ and $\Gamma^{r^{s}}_{\underline{X}}(\mathcal{I}^{\mathbb{P}})$, thereby linking ideal structure to probabilistic convergence behavior. Collectively, this work advances the theory of probabilistic convergence under ideals and elucidates the topological and combinatorial structure of rough limit and cluster sets in stochastic settings.
Abstract
Given a probability space $(S,Δ, \mathbb{P})$ and a separable metric space $(U,d)$, the $Ky~Fan$ metric $ρ(X,Y)$ 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 $ρ(X,Y)=\inf \{\varepsilon>0:\mathbb{P}(d(X,Y)>\varepsilon)\leq\varepsilon\}.$ 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 $ρ$, and that, for a certain class of ideals, it forms an $F_{σδ}$ 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 ($\mathfrak{X}^0,ρ)$. 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.