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.


1 Answer 1


Let $X_i \sim_{\text{i.i.d}} \text{LNorm}(\mu,\,\sigma)$ with the meaning that the r.v. $\log X_i$ is normal with mean $\mu$ and standard deviation $\sigma$. Considering $M_n := \max_{1 \leqslant i \leqslant n} X_i$, we know that there exist two sequences $a_n >0$ and $b_n$ such that

$$ \tag{1} \frac{M_n - b_n}{a_n} \to \text{Gum}(0, 1) $$

where $\text{Gum}(\nu,\,\beta)$ denotes the Gumbel distribution with location $\nu$ and scale $\beta$. This means that $F_{M_n}(a_n x + b_n) \to F_{\text{Gum}}(x;\,0,\,1)$ for all $x$.

Quite obviously the two sequences $a_n$ and $b_n$ depend on $\mu$ and $\sigma$, so they could be denoted as $a_n(\mu,\,\sigma)$ and $b_n(\mu,\,\sigma)$. For instance if $\mu$ is replaced by $\mu +1$ then the distribution of $X_i$ is replaced by that of $e X_i$ and the distribution of $M_n$ is replaced by that of $e M_n$, implying that $a_n$ and $b_n$ have to be replaced by $ea_n$ and $eb_n$ to maintain the same limit. Similarly if we replace $\mu$ by $0$ with $\sigma$ unchanged, $X_i$ is to be replaced by $e^{-\mu} X_i$ and then $a_n$ and $b_n$ must be replaced by $e^{-\mu} a_n$ and $e^{-\mu}b_n$.

The question can be formulated as: if we use the sequences $a_n(0, 1)$ and $b_n(0, \,1)$ at the left-hand side of (1) - instead of the due $a_n(\mu,\,\sigma)$ and $b_n(\mu,\,\sigma)$ - do we get $\text{Gum}(\mu,\,\sigma)$ at the right-hand side? The answer is then no, because the parameters of the Gumbel are indeed location and scale parameters, while this is not true for the log-normal. The parameter $\sigma$ of the log-normal impacts the tail, as can be seen by the fact that the coefficient of variation increases with $\sigma$. While $\text{LNorm}(\mu,\,\sigma)$ always remains in the Gumbel domain of attraction, the sequences $a_n$ and $b_n$ must tend to $\infty$ more rapidly as $\sigma$ increases. It can be proved that we can in (1) use sequences $a_n$ and $b_n$ such that $$ b_n(\mu, \sigma) = e^\mu \, b_n(0, 1)^\sigma, \qquad a_n(\mu, \sigma) = \sigma \,(2 \log n)^{-1/2} b_n(\mu,\,\sigma), $$ see Embrechts P., Klüppelberg C. and Mikosch T. table 3.4.4 pp 155-157. If we use sequences $a_n$ and $b_n$ with a wrong $\sigma$, we will not get a non-degenerate limit for the left-hand side of (1), because the growth rates of $a_n$ and $b_n$ are then unsuitable for the tail of $X_i$.


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