A Markov decision process with delayed state availability

We consider a finite horizon Markov decision process with delayed state availability. We show that at state t, a sufficient statistic is the state of the system at stage t-k and all actions taken from stage t-k through stage t-1, given that the state of the system at stage t-k is the most recent state value made available. Further, we present a bound that relates the value functions of the k-delay and the (k-1)-delay cases