A law of large numbers for upcrossing measures
This paper develops a law of large numbers for a measure-valued process counting the number of upcrossings of intervals \(\{(x,y) : x < y\}\) by a class of random processes. This is motivated by fatigue analysis (the analysis of the fatigue in a mechanical part in response to strain). Empirical evidence suggests that the degree of fatigue experienced by a part is strongly related to the number of times the strain/stress function traverses upwards across the intervals \(\{(x,y) : x < y\}\). Thus estimation of the upcrossing measure for the strain function becomes an important practical question. Variants of this measure are known in engineering as the rainflow or range-pair-range statistic. Let \(X(t)\), \(t \in R\), be a cadlag stationary process and let \(u(X,x,y)\) denote the number of upcrossings of the interval \((x,y)\) by \(X\) in the period \([0,T]\). The paper describes a discrete random measure \(\nu\) on \(\Delta = \{(x,y) \in R^ 2 : x < y\}\) such that \(\nu_{[0,T]} (X, (- \infty,x] \times [y,\infty)) = u(X,x,y)\), and proves that as \(T \to \infty\), \(T^{-1} \nu_{[0,T]} (X, \cdot)\) converges almost surely in the vague topology to a Radon measure, \(\widetilde \mu\). For ergodic processes, \(\widetilde \mu\) is deterministic. Chapter 2 provides rigorous definitions of the upcrossings and related measures for deterministic \(X\), and establishes useful additivity properties. Chapter 3 states and proves the desired law of large numbers. The proof uses Kingman's subadditive ergodic theorem and the previous additivity properties. Chapter 4 presents 5 examples where \(\widetilde \mu\) is calculated. These include i.i.d. variables, diffusion processes, and continuous local martingales.
scientific article; zbMATH DE number 713907
scientific article