= Solution
This is the \b[<generalized Pareto distribution>] for the excess $X-u$, with scale $\sigma>0$ and shape $\xi$. Its support requires $1+\xi(x-u)/\sigma>0$. The <Pickands-Balkema-de Haan theorem> says that, for distributions in an appropriate extreme-value domain of attraction, the conditional distribution of excesses above a sufficiently high threshold approaches a generalized Pareto form. This makes it the natural <peaks-over-threshold method> model; it is an asymptotic justification, not an assertion that every threshold is sufficiently high.
For $\xi>0$, the tail decays as a power and has no finite upper endpoint, giving a heavy tail. For $\xi=0$, the limiting distribution is exponential, with survival probability $\exp\{-(x-u)/\sigma\}$. For $\xi<0$, there is a finite upper endpoint $u-\sigma/\xi$, giving a bounded tail. Thus the sign of the <shape parameter> distinguishes heavy, exponential-type and bounded tails.
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