SummaryConsider a discrete time Markov chainX(n) denned on {0,1,…} and let P be the transition probability matrix governingX(n).This paper shows that, if a transformed matrix of P is totally positive of order 2, thenpoj(n) andpio(n) are unimodal with respect to n, wherepij(n) = Pr[X(n) = j |X(0) = i]. Furthermore, the modes ofpoj(n) andpio(n) are non‐increasing injand I, respectively, when additionally P itself is totally positive of order 2. These results are transferred to a class of semi‐Markov processes via a uniformiz