A Tauberian theorem for ideal statistical convergence

Abstract Given an ideal I on the positive integers, a real sequence ( x n ) is said to be I -statistically convergent to l provided that n ∈ N : 1 n | { k ≤ n : x k ∉ U } | ≥ e ∈ I for all neighborhoods U of l and all e > 0 . First, we show that I -statistical convergence coincides with J -convergence, for some unique ideal J = J ( I ) . In addition, J is Borel [analytic, coanalytic, respectively] whenever I is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if I is the summable ideal { A ⊆ N : ∑ a ∈ A 1 ∕ a ∞ } or the density zero ideal { A ⊆ N : lim n → ∞ 1 n | A ∩ [ 1 , n ] | = 0 } then 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 I is maximal.