# A Tauberian theorem for ideal statistical convergence

## TL;DR

This paper explores a generalized form of statistical convergence based on ideals, establishes its equivalence with a specific type of convergence, and extends classical Tauberian theorems to this new framework.

## Contribution

It introduces a novel connection between ideal-based statistical convergence and classical convergence, including a unique ideal correspondence and conditions for equivalence.

## Key findings

- $ ext{I}$-statistical convergence coincides with $ ext{J}$-convergence for a unique ideal $ ext{J}$.
- When $ ext{I}$ is the summable or density zero ideal, $ ext{I}$-statistical convergence equals classical statistical convergence.
- Maximal ideals do not allow $ ext{I}$-statistical convergence to coincide with statistical convergence.

## Abstract

Given an ideal $\mathcal{I}$ on the positive integers, a real sequence $(x_n)$ is said to be $\mathcal{I}$-statistically convergent to $\ell$ provided that $$ \textstyle \left\{n \in \mathbf{N}: \frac{1}{n}|\{k \le n: x_k \notin U\}| \ge \varepsilon\right\} \in \mathcal{I} $$ for all neighborhoods $U$ of $\ell$ and all $\varepsilon>0$. First, we show that $\mathcal{I}$-statistical convergence coincides with $\mathcal{J}$-convergence, for some unique ideal $\mathcal{J}=\mathcal{J}(\mathcal{I})$. In addition, $\mathcal{J}$ is Borel [analytic, coanalytic, respectively] whenever $\mathcal{I}$ is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if $\mathcal{I}$ is the summable ideal $\{A\subseteq \mathbf{N}: \sum_{a \in A}1/a<\infty\}$ or the density zero ideal $\{A\subseteq \mathbf{N}: \lim_{n\to \infty} \frac{1}{n}|A\cap [1,n]|=0\}$ then $\mathcal{I}$-statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if $\mathcal{I}$ is maximal.