Some Remarks on a General Class of Markov Processes with Discrete Time Parameter and Dependent Increments

In this paper we consider a general class of Markov processes with discrete time parameter {X _n : n = 0, 1, 2, …}, where the transitions from one state to another are determined by:

The functions h _j (·) are strictly increasing, and in general the Markov processes have dependent increments. In section 1 the model is described more precisely. Considering the sequence of distribution functions F _n(x) = Pr[X _n ≤ x] (n = 0, 1, 2, ….) a numerical method is given by which upper and lower bounds can be determined for F _n(x) (n ≥ n _o). Numerical results are given for some special cases, and the determination of an alarm level in a flame failure control system is described as an application. The results show that the upper and lower bounds can lie very close together.