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May 2016 Local limit theorems for shock models
Edward Omey, Rein Vesilo
Braz. J. Probab. Stat. 30(2): 221-247 (May 2016). DOI: 10.1214/14-BJPS274


In many physical systems, failure occurs when the stress after shock $n$ first exceed a critical level $x$. We consider the number of shocks $\tau(x)$ to failure and obtain more detailed information that is usually obtained about asymptotic distribution by using local limit theorems. We consider extreme and cumulative shock models with both univariate and multivariate shock types. We derive the limiting distribution of $\tau(x)$ and the rate of convergence to that. For the extreme shock model, rate of convergence for regularly varying shock distributions is found using the weighted Kolmorogov probability metric. For the cumulative shock model, we examine the rate of convergence to Gaussian densities.


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Edward Omey. Rein Vesilo. "Local limit theorems for shock models." Braz. J. Probab. Stat. 30 (2) 221 - 247, May 2016.


Received: 1 November 2013; Accepted: 1 December 2014; Published: May 2016
First available in Project Euclid: 31 March 2016

zbMATH: 1341.60107
MathSciNet: MR3481102
Digital Object Identifier: 10.1214/14-BJPS274

Keywords: Extreme value theory , local limit theory , regular variation , renewal theory , shock models

Rights: Copyright © 2016 Brazilian Statistical Association


Vol.30 • No. 2 • May 2016
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