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 $X_i$.
