Weak statistical convergence and weak filter convergence for unbounded sequences

For every weakly statistically convergent sequence (xn) with increasing norms in a Hilbert space we prove that . This estimate is sharp. We study analogous problem for some other types of weak filter convergence, in particular for the Erdös–Ulam filters, analytical P-filters and Fσ filters. We present also a refinement of the recent Aron–Garcia–Maestre result on weakly dense sequences that tend to infinity in norm.