## Brazilian Journal of Probability and Statistics

### Local limit theorems for shock models

#### Abstract

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.

#### Article information

Source
Braz. J. Probab. Stat., Volume 30, Number 2 (2016), 221-247.

Dates
Accepted: December 2014
First available in Project Euclid: 31 March 2016

https://projecteuclid.org/euclid.bjps/1459429711

Digital Object Identifier
doi:10.1214/14-BJPS274

Mathematical Reviews number (MathSciNet)
MR3481102

Zentralblatt MATH identifier
1341.60107

#### Citation

Omey, Edward; Vesilo, Rein. Local limit theorems for shock models. Braz. J. Probab. Stat. 30 (2016), no. 2, 221--247. doi:10.1214/14-BJPS274. https://projecteuclid.org/euclid.bjps/1459429711

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