  We prove that there exists a martingale $f\in H_p $ such that the subsequence
$\{L_{2^n}f \}$ of N\"orlund logarithmic means with respect to the Walsh system
are not bounded in the Lebesgue space $weak-L_p $ for $0<p<1 $. Moreover, we
prove that for any $f\in L_p(G),$ $p\geq 1, $ $L_{2^n}f$ converge to $f$ at any
Lebesgue point $x$. Some new related inequalities are derived.
