A Tauberian theorem for ideal statistical convergence

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
Comment: 15 pages, comments are welcome