Moments of Transition-Additive Random Variables Defined on Finite, Regenerative Random Processes

Random processes consisting of a sequence of jumps from one to another of a finite set of states are considered. Such processes are regenerative if the progress after the nth jump depends only upon the state entered at the jump. Examples include discrete and continuous Markov processes. A method is given to restrict attention to a subset of sample paths according to criteria based on the transitions allowed for a single jump. The primary concern is with transition-additive random variables: these sum for any valid sample path the values of independent random variables assigned to each jump in the path. A simple formula for finding all moments of such random variables is derived. Necessary and sufficient conditions for the existence of the solutions are demonstrated, and illustrations of the computational simplicity of the approach are provided. ‘1 1988 Academz Press. Inc