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Local limit theorems for shock models

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  • Omey, Edward

    () (Hogeschool-Universiteit Brussel (HUB), Belgium)

  • Vesilo, Rein

    () (Macquarie University, Dept. of Electronics, Sydney, Australia)

Abstract

In this paper we study the local behaviour of a characteristic of two types of shock models. In many physical systems, a failure occurs when the stress or the fatigue, represented by $\epsilon(n)$, reaches a critical level $x$. We are interested in the time $\tau(x)$ for which this happens for the first time. In the cumulative shock model we assume that $\epsilon(n) = \sum_{i=1}^n X_i$ is an acummulation of independent shocks $\Ksi_i$. In the extreme shock model, we assume that $\epsilon(n) = \max(X_1,X_2, ..., X_n)$ where the damage to the system is measured in terms of the largest shock up to now. For both models we.obtain a local limit theorem for the corresponding time $\tau(x)$.

Suggested Citation

  • Omey, Edward & Vesilo, Rein, 2011. "Local limit theorems for shock models," Working Papers 2011/23, Hogeschool-Universiteit Brussel, Faculteit Economie en Management.
  • Handle: RePEc:hub:wpecon:201123
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    References listed on IDEAS

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    1. Omey, E. & Rachev, S. T., 1991. "Rates of convergence in multivariate extreme value theory," Journal of Multivariate Analysis, Elsevier, vol. 38(1), pages 36-50, July.
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    Keywords

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

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