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In this paper we derive characterizations of boundedness of the subsequencesof partial sums with respect to Vilenkin system on the martingale Hardy spaceswhen $ 0<p<1 $. Moreover, we find necessary and sufficient conditions for themodulus of continuity of $f\in H_{p}$ martingales, which provide convergence ofsubsequences of partial sums on the martingale Hardy spaces. It is also provedthat these results are the best possible in a special sense. As applications,both some well-known and new results are pointed out.

On the convergence of partial sums with respect to Vilenkin system on the martingale Hardy spaces