Fragility Distributions and their approximations

Given a sequence of $n$ identically distributed random variables with common distribution $F$, the \emph{fragility distribution of order $m$}, represented by $\text{FD}$, is the limit conditional distribution of the number of exceedances given there are at least $m$ exceedances, as the threshold tends to the right end point of $F$. In this paper we are concerned with the existence of $\text{FD}$ and its asymptotic behaviour when $n$ becomes large. For a stationary sequence with its exceedance process converging to a compound Poisson process, we derive an explicit formula for calculating ${\lim }_{n\to \mathrm{\infty }}\text{FD}$. We also establish Stein’s method for estimating the errors involved in fragility distribution approximations.