A law of large numbers for upcrossing measures
type: Item
instance of: scholarly article
title: A law of large numbers for upcrossing measures
zbMATH Open document ID: 0813.60032
DOI: 10.1016/0304-4149(94)00002-6
author: Michael Scheutzow
published in: Stochastic Processes and their Applications
publication date: 1995-05-30T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 713907
zbMATH Keywords: fatigue analysis
zbMATH Keywords: rainflow counting
zbMATH Keywords: upcrossings
zbMATH Keywords: local martingales
reviewed by: Andre R. Dabrowski
MaRDI profile type: Publication
full work available at URL: 0304-4149(94)00002-6
OpenAlex ID: W2027805078
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description: scientific article; zbMATH DE number 713907