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.