Weak statistical convergence and weak filter convergence for unbounded sequences

For every weakly statistically convergent sequence ( x n ) with increasing norms in a Hilbert space we prove that sup n ‖ x n ‖ / n < ∞ . 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 F σ filters. We present also a refinement of the recent Aron–Garcia–Maestre result on weakly dense sequences that tend to infinity in norm.