Pickands–Balkema–De Haan theorem

Remarks

Idea of the proof. Membership in the domain of attraction of $H_\xi$ is equivalent to the existence of a positive function $a(u)$ with $\frac{1 - F(u + a(u) y)}{1 - F(u)} \to (1 + \xi y)^{-1/\xi}$ as $u \uparrow x_F$, for each admissible $y$. The left side is $1 - F_u(a(u) y)$, so this is pointwise convergence of rescaled excess laws to $G_{\xi, 1}$, and $\beta(u) = a(u)$ works. Uniformity in $y$ follows because the functions are monotone and the limit is continuous.

Examples. For the exponential law $F(x) = 1 - e^{-x}$, $F_u(y) = 1 - e^{-y}$ for every $u$, so $\xi = 0$ and $\beta(u) = 1$ give equality. For the Pareto law $1 - F(x) = x^{-\alpha}$ for $x \ge 1$, $F_u(y) = 1 - (1 + y/u)^{-\alpha} = G_{1/\alpha, \, u/\alpha}(y)$ for $u \ge 1$. For the uniform law on $[0, 1]$, $F_u(y) = y/(1 - u) = G_{-1, \, 1 - u}(y)$.

The hypothesis is not automatic: some laws lie in no domain of attraction, for example the Poisson and geometric laws. Their excesses over high thresholds have no generalised Pareto limit.

Going deeper

For a student

The theorem is the threshold counterpart of the Fisher–Tippett–Gnedenko theorem, which describes the limit laws of maxima. The same shape parameter $\xi$ appears in both. One has $\xi > 0$ for heavy, Pareto-like tails, $\xi = 0$ for light tails such as the normal and exponential laws, and $\xi < 0$ for laws with a finite right endpoint. In practice it justifies the peaks-over-threshold method of extreme value statistics. One fits $G_{\xi, \beta}$ to the excesses over a high threshold by maximum likelihood and uses the fit to estimate high quantiles and tail probabilities.