Properties of locally convergent sequences with respect to median filter

In this paper, for infinite-length sequences (or sequences), we prove that a sequence is convergent with respect to the median filter with window width 2k+12k+1 if and only if the sequence is locally convergent on a segment of length 2k−12k−1 in the sequence. Moreover, the length 2k−12k−1 is minimal for k≠2,3k≠2,3.