Limit theorems for cumulative processes

Abstract

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 the integral of the regenerative process over one cycle and the supremum of the absolute value of this integral over all possible initial segments of a cycle. The tail behavior of the distribution of the second random variable determines whether the cumulative process obeys the same limit theorem as the partial sums of the cycle integrals. Interesting open problems are the necessary conditions for the weak law of large numbers and the ordinary central limit theorem.