The weights of weighted median and recursive weighted median filters have close relationship with the root signal characteristics of these filters. In this paper, we find a set of conditions on weights under which a recursive weighted median filter preserves monotone or locally monotone sequence and that under which any input sequence converges to a locally monotone sequence after a finite number of passes. We also find a set of conditions on weights under which the output of a recursive weighted median filter is the same as that of a recursive median filter.