Extreme Value Theory
4. Block maxima results – examples of
different limiting behaviour
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The limiting behaviours of many traditional distributions
are well-known. For example, block maxima values for the normal, log-normal and exponential
distributions converge to the Gumbel
distribution. Those for the Student’s t,
inverse gamma,
F and Pareto
distributions converge to the Fréchet
distribution.
Two examples where it is easy to derive the limiting
distributions are the exponential and the Pareto distribution:
Exponential
Suppose the underlying distribution is an exponential
distribution with distribution function 
for
and
.
Then by choosing normalising sequences 
and
we
have:
So 
.
Pareto
Suppose the underlying distribution is a Pareto distribution
with
distribution function 
for
and
.
Then by choosing the normalising sequences 
and
we
have:
So 
.
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