Powers and Products of Regenerative Phenomena

Abstract

SummaryThe simultaneous occurrence of two independent regenerative phenomena defines a third, whose p‐function is the product of the first two. Thus integer powers of p‐functions are p‐functions. The corresponding result for non‐integer powers (with exponent α > 1) has been proved for discrete time phenomena and for standard continuous time phenomena. There are still open questions, notably whether the class of Markov p‐functions is closed under non‐integer powers. These questions are addressed by means of a new technique which relates the atoms of the canonical measure to ‘kinks’ in the p‐function. This provides new information even for products of p‐functions.