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 2 k + 1 if and only if the sequence is locally convergent on a segment of length 2 k − 1 in the sequence. Moreover, the length 2 k − 1 is minimal for k ≠ 2 , 3 .