## Brazilian Journal of Probability and Statistics

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

### 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**

Received: November 2013

Accepted: December 2014

First available in Project Euclid: 31 March 2016

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/14-BJPS274

**Mathematical Reviews number (MathSciNet)**

MR3481102

**Zentralblatt MATH identifier**

1341.60107

**Keywords**

Renewal theory shock models regular variation extreme value theory local limit theory

#### 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