Asymptotic Properties of Cumulative Processes

Cumulative processes were defined and explored by W. L. Smith [3]. Smith derived an analogue of the elementary renewal theorem for cumulative processes. We derive analogues of the Blackwell and key renewal theorems for a subclass of cumulative processes which we call strongly cumulative. This subclass is defined in a manner similar to Smith’s definition of a regenerative process [4], and appears to include all standard examples of cumulative processes. We also investigate processes of the form ( t ) = ∫₀ᵗ V( s )ds, where V is a regenerative process. Smith has shown under weak conditions that $Y(t)/t$ converges a.s. and in expectation to κ ₁ /μ ₁ where μ ₁ is the expected interarrival time of the embedded renewal process and κ ₁ is the expected change in Y over a renewal cycle. We show that κ ₁ /μ ₁ is equal to the mean of the limiting distribution of $V( t )$.