ON THE UNIMODALITY OF TRANSITION PROBABILITIES IN MARKOV CHAINS

Consider a discrete time Markov chain X(n) denned on {0,1,…} and let P be the transition probability matrix governing X(n). This paper shows that, if a transformed matrix of P is totally positive of order 2, then poj(n) and pio(n) are unimodal with respect to n, where pij(n) = Pr[X(n) = j |X(0) = i]. Furthermore, the modes of poj(n) and pio(n) are non-increasing in j and 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 uniformization.