On the transition from a Markov chain to a continuous time process

AbstractStarting from a real-valued Markov chain X0,X1,…,Xn with stationary transition probabilities, a random element {Y(t);t∈[0, 1]} of the function space D[0, 1] is constructed by letting Y(kn)=Xk, k= 0,1,…,n, and assuming Y (t) constant in between. Sample tightness criteria for sequences {Y(t);t∈[0,1]};n of such random elements in D[0, 1] are then given in terms of the one-step transition probabilities of the underlying Markov chains. Applications are made to Galton-Watson branching processes