ASYMPTOTIC PROPERTIES OF CUMULATIVE PROCESSES
CALIFORNIA UNIV BERKELEY OPERATIONS RESEARCH CENTER
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The theory of cumulative processes, introduced and developed by W. L. Smith, provides a significant generalization of the renewal counting process. One examines the question of extending the Blackwell and key renewal theorems to cumulative processes. For a subclass of cumulative processes, which one calls strongly cumulative, the Blackwell and key renewal theorems hold under very general conditions. This class of cumulative processes includes all the standard examples of cumulative processes. One also studies processes of the form Yt the integral from o to t Vsds where V is a regenerative process. Smith has shown that under mild conditions Vs converges in distribution, say to Vinfinity, as s approaches infinity, and that Ytt converges almost surely and in expectation to Kappa sub 1mu sub 1 a constant. The result is that Kappa sub 1mu sub 1 EVinfinity. This holds even if lim as t approaches infinity, of EVt does not exist.
- Operations Research