We show that sufficient conditions in terms of moments for cumulative processes (additive functionals of regenerative processes) to satisfy the central limit theorem and the weak law of large numbers established in Glynn and Whitt (Stochastic Process. Appl. 47 (1993) 299-314) are also necessary,...

Necessary and sufficient conditions are established for cumulative process (associated with regenerative processes) to obey several classical limit theorems; e.g., a strong law of large numbers, a law of the iterated logarithm and a functional central limit theorem. The key random variables are...

To help provide a theoretical basis for approximating queues with superposition arrival processes, we prove limit theorems for the queue-length process in a [Sigma] GIi/G/s model, in which the arrival process is the superposition of n independent and identically distributed stationary renewal...