A law of large numbers for upcrossing measures

Abstract

AbstractWe present a mathematical treatment of the so called RFC-counting which is applied to functions from subsets of R to R and which essentially counts upcrossings for each pair of levels. In mechanical engineering it is applied to stress or strain histories to assess their potential fatigue damage. We associate three measures on R2 with RFC-counting and study their properties. Using the subadditive ergodic theorem of Kingman (1975) we prove a law of large numbers for these measures when they are applied to the paths of a stationary process. We compute the limit μ̃ explicitly e.g. for one-dimensional stationary diffusion processes. μ̃ may be compared with the spectral measure