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Lognormal distribution belongs to the Gumbel maximum domain of attraction, where:

$F^{logN}(x; \mu,\sigma)=\Phi\left(\frac{\ln x - \mu}{\sigma}\right)$,

$F^{Gum}(x;\mu,\beta) = e^{-\exp\left({-\frac{x-\mu}{\beta}}\right)}$

My question: Do we have $\mu=\mu$ and $\sigma=\beta$ ?

The Generalized Extreme Value distribution also uses notation $\beta=\sigma$ (Gumbel is the limit case $\xi =0$), and comparing the CDFs for Standard-Lognormal and Standard-Gumbel would again imply the parameters coincide. But I am not sure about it, because Gumbel is a limiting case of Lognormal Maxima, so there might be some transformation of the parameters aswell.

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