Pickands–Balkema–De Haan theorem
The Pickands–Balkema–De Haan theorem gives the asymptotic tail distribution of a random variable, when its true distribution is unknown. It is often called the second theorem in extreme value theory. Unlike the first theorem (the Fisher–Tippett–Gnedenko theorem), which concerns the maximum of a sample, the Pickands–Balkema–De Haan theorem describes the values above a threshold. The theorem owes its name to mathematicians James Pickands, Guus Balkema, and Laurens de Haan. Conditional excess distribution function
For an unknown distribution function  of a random variable , the Pickands–Balkema–De Haan theorem describes the conditional distribution function  of the variable  above a certain threshold . This is the so-called conditional excess distribution function, defined as

 

for , where  is either the finite or infinite right endpoint of the underlying distribution . The function  describes the distribution of the excess value over a threshold , given that the threshold is exceeded. Statement
Let  be the conditional excess distribution function. Pickands, Balkema and De Haan posed that for a large class of underlying distribution functions , and large ,  is well approximated by the generalized Pareto distribution, in the following sense. Suppose that there exist functions , with  such that  as  converge to a non-degenerate distribution, then such limit is equal to the generalized Pareto distribution:

 ,

where
,  if 
,  if 

Here σ > 0, and y ≥ 0 when k ≥ 0 and 0 ≤ y ≤ −σ/k when k < 0. These special cases are also known as  

Exponential distribution with mean ,  if k = 0,
Uniform distribution on ,  if k = -1,
 Pareto distribution, if k > 0. The class of underlying distribution functions  are related to the class of the distribution functions  satisfying the Fisher–Tippett–Gnedenko theorem. Since a special case of the generalized Pareto distribution is a power-law, the Pickands–Balkema–De Haan theorem is sometimes used to justify the use of a power-law for modeling extreme events. The theorem has been extended to include a wider range of distributions. While the extended versions cover, for example the normal and log-normal distributions, still continuous distributions exist that are not covered. See also 

 Stable distribution

References 

Probability theorems
Extreme value data
Tails of probability distributions