EXAMPLE 2.2.9. (Maxima of Cauchy random variables). Let X be a sequence of iid standard Cauchy random variables, i.e. the density function is given by for . Using l’Hospitals rule we obtain
Then as . Hence, for as and
for . Hence F belongs to the maximum domain of attraction of the Frèchet [sic] distribution. The normalizing constants can be chosen to be and
EXAMPLE 2.2.10. (Maxima of exponential random variables). Let X be a sequence of iid standard exponential random variables, i.e., the distribution function F of X is given by for Then,
Hence, F belongs to the maximum domain of attraction of the Gumbel distribution. The normalizing constants can be chosen to be and
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Example 3.2 (Maxima of exponential random variables). Let (Xk) be a sequence of iid standard exponential random variables, i.e. the distribution function F of Xk is given by for Then
Hence, F belongs to the maximum domain of attraction of the Gumbel distribution . The normalizing constants can be chosen to be and .
Example 3.3 (Maxima of Cauchy random variables). Let (Xk) be a sequence of iid standard Cauchy random variables, i.e. the density function f of Xk is given by
for . Using l’Hospitals rule we obtain
i.e. as . Hence, for as and
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For . Hence, F belongs to the maximum domain of attraction of the Fréchet distribution . The normalizing constants can be chosen to be and .
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