LIMIT-THEOREMS FOR CUMULATIVE PROCESSES

Necessary and sufficient conditions are established for cumulative pro cess (associated with regenerative processes) to obey several classica l limit theorems; e.g., a strong law of large numbers, a law of the it erated logarithm and a functional central limit theorem. The key rando m variables are the integral of the regenerative process over one cycl e and the supremum of the absolute value of this integral over all pos sible initial segments of a cycle. The tail behavior of the distributi on of the second random variable determines whether the cumulative pro cess obeys the same limit theorem as the partial sums of the cycle int egrals. Interesting open problems are the necessary conditions for the weak law of large numbers and the ordinary central limit theorem.