arXiv subjects

Sanjoy Ghosal

Publications and source records attributed to Sanjoy Ghosal.

5 recordsLinked to original sources

arXiv2026

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

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.

math.PR↗

arXiv2025

Rough Weighted Ideal Convergence and Korovkin-Type Approximation via weighted equi-ideal convergence

If $\omega_t > \beta$ for every $t \in \mathbb{N}$ and for some $\beta > 0$, then the sequence $\{\omega_t\}_{t \in \mathbb{N}}$ represents a weighted sequence of real numbers. In this article, we primarily introduce the concepts of rough weighted ideal limit set and rough weighted ideal cluster points set associated with sequences in normed spaces. Building on these concepts, we derive several important results, including a characterization of maximal ideals, a representation of closed sets in normed spaces, and an analysis of the minimal convergent degree required for the rough weighted ideal limit set to be non-empty. Furthermore, we demonstrate that for an analytic $P$-ideal, the rough weighted ideal limit set forms an $F_{\sigma\delta}$ subset of the normed space. Finally, we introduce the concept of weighted equi-ideal convergence for sequences of functions with respect to analytic $P$-ideals, extending the notion of equi-statistical convergence [Balcerzak et al., J. Math. Anal. Appl. {328} (1) (2007)]. As an application of this notion, we establish a Korovkin-type approximation theorem that serves both as a generalization of [Theorem 2.4, Karaku{\c{s}} et al., J. Math. Anal. Appl. {339} (2) (2008)] and a correction to [Theorem 2.2, Akda\u{g}, Results Math. {72} (3) (2017)].

math.FA↗

arXiv2021

When degree of roughness is a neighborhood over locally solid Riesz spaces

In this paper we introduce the notion of rough weighted $\mathcal{I}_τ$-limit points set and weighted $\mathcal{I}_τ$-cluster points set in a locally solid Riesz space which are more generalized version of rough weighted $\mathcal{I}$-limit points set and weighted $\mathcal{I}$-cluster points set in a $θ$-metric space respectively. Successively to compare with the following important results of Fridy [Proc. Amer. Math. Soc. {118} (4) (1993), 1187-1192] and Das [Topology Appl. {159} (10-11) (2012), 2621-2626], respectively be stated as \begin{description} \item[(i)] Any number sequence $x=\{x_{n}\}_{n\in \mathbb{N}},$ the statistical cluster points set of $x$ is closed, \item[(ii)] In a topological space the $\mathcal{I}$-cluster points set is closed, \end{description} we show that in general, the weighted $\mathcal{I}_τ$-cluster points set in a locally solid Riesz space may not be closed. The resulting summability method unfollows some previous results in the direction of research works of Aytar [Numer. Funct. Anal. Optim. {29} (3-4) (2008) 291-303], D$\ddot{\mbox{u}}$ndar [Numer. Funct. Anal. Optim. {37} (4) (2016) 480-491], Ghosal [Math. Slovaca {70} (3) (2020) 667-680] and Savaş, Et [Period. Math. Hungar. 71 (2015) 135-145].

math.GN↗

arXiv2016

A notion of $αβ$-statistical convergence of order $γ$ in probability

A sequence of real numbers $\{x_{n}\}_{n\in \mathbb{N}}$ is said to be $αβ$-statistically convergent of order $γ$ (where $0<γ\leq 1$) to a real number $x$ \cite{a} if for every $δ>0,$ $$\underset{n\rightarrow \infty} {\lim} \frac{1}{(β_{n} - α_{n} + 1)^γ}~ |\{k \in [α_n,β_n] : |x_{k}-x|\geq δ\}|=0.$$ where $\{α_{n}\}_{n\in \mathbb{N}}$ and $\{β_{n}\}_{n\in \mathbb{N}}$ be two sequences of positive real numbers such that $\{α_{n}\}_{n\in \mathbb{N}}$ and $\{β_{n}\}_{n\in \mathbb{N}}$ are both non-decreasing, $β_{n}\geq α_{n}$ $\forall ~n\in \mathbb{N},$ ($β_{n}-α_{n})\rightarrow \infty$ as $n\rightarrow \infty.$ In this paper we study a related concept of convergences in which the value $|x_{k}-x|$ is replaced by $P(|X_{k}-X|\geq \varepsilon)$ and $E(|X_{k}-X|^{r})$ repectively (Where $X, X_k$ are random variables for each $k\in \mathbb{N}$, $\varepsilon>0$, $P$ denote the probability, $E$ denote the expectation) and we call them $αβ$-statistical convergence of order $γ$ in probability and $αβ$-statistical convergence of order $γ$ in $r^{\mbox{th}}$ expectation respectively. The results are applied to build the probability distribution for $αβ$-strong $p$-Ces$\grave{\mbox{a}}$ro summability of order $γ$ in probability and $αβ$-statistical convergence of order $γ$ in distribution. Our main objective is to interpret a relational behavior of above mentioned four convergences.

math.PR↗

arXiv2016

Statistical convergence of order $α$ in probability

In this paper ideas of different types of convergence of a sequence of random variables in probability, namely, statistical convergence of order $α$ in probability, strong $p$-Ces$\grave{\mbox{a}}$ro summability of order $α$ in probability, lacunary statistical convergence or $S_θ$-convergence of order $α$ in probability, ${N_θ}$-convergence of order $α$ in probability have been introduced and their certain basic properties have been studied.

math.PR↗