  We investigate the subsequence $\{t_{2^n}f \}$ of N\"{o}rlund means with
respect to the Walsh system generated by non-increasing and convex sequences.
In particular, we prove that a big class of such summability methods are not
bounded from the martingale Hardy spaces $H_p$ to the space $weak-L_p $ for
$0<p<1/(1+\alpha) $, where $0<\alpha<1$. Moreover, some new related
inequalities are derived. As application, some well-known and new results are
pointed out for well-known summability methods, especially for N\"{o}rlund
logarithmic means and Ces\`aro means.
