# Gamma distribution Norming constant for extreme minima

the norming constants for extreme maxima of Gamma distribution is known and is give in link.springer.com/article/10.1007/s10687-010-0125-3. I would like to know is there reference or paper that states the norming constant for the extreme minima. The exact problem is stated below:

Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be $X_n=min(a_1,a_{2},\cdots, a_n)$, where $a_i$s are Gamma random variables. It is well-known in extreme value theory that the CDF of $X_n$ and $Y_n$ converges (in distribution) as follows:

$$\lim_{n\rightarrow \infty}~~~Pr\left(\frac{Y_n-\mu}{\sigma}\leq x\right)\rightarrow G_M(x)~~~~~~~~~~(P1)$$

$$\lim_{n\rightarrow \infty}~~~Pr\left(\frac{X_n-\mu_{1}}{\sigma_{1}}\leq x\right)\rightarrow W_m(x)~~~~~~~~~~(P2)$$

where $G_M(x)$ is the Gumbel and $W_m(x)$ is the Weibull CDFs for maxima and minima respectively and the values of $\mu$ and $\sigma$ can be given explicitly (see for example link.springer.com/article/10.1007/s10687-010-0125-3).

My question is: Is there a literature that provides the norming constants $\mu_1$ and $\sigma_1$ for the minima? will appreciate your answer and possible references.

I have found answer for my own question. The construction of norming constants $\sigma_1$ and $\mu_1$ for the minima can be found from the following references.