I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem.
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 Chi-square random variables with $K$ degrees of freedom. It is well-known in extreme value theory that the CDF of $X_n$ and $Y_n$ converges (in distribution) as follows:
$$\lim_{n\rightarrow \infty}~~~Pr\left(\frac{Y_n-\mu}{\sigma}\leq x\right)\rightarrow G_M(x)~~~~~~~~~~(P1)$$
$$\lim_{n\rightarrow \infty}~~~Pr\left(\frac{X_n-\mu_{1}}{\sigma_{1}}\leq x\right)\rightarrow G_m(x)~~~~~~~~~~(P2)$$
where $G_M(x)$ and $G_m(x)$ are Gumbel 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: Are the normalizing constants same for both the problem $P1$ and $P2$? I mean, is it true that $\mu=\mu_1$ and $\sigma=\sigma_1$? I could not find the values of $\mu_1$ and $\sigma_1$ in the literature.I will appreciate your answer and possible references.
I am answering my own question above. The correct answer is that $\mu \neq \mu_1$ and $\sigma \neq \sigma_1$. Thus, the normalizing constants for the maximum and minimum extreme value distributions are entirely different.
This is illustrated in the monograph "Extreme Value and Related Models with Applications in Engineering and Science by Enrique Castillo, Ali S. Hadi, N. Balakrishnan, Jose M. Sarabia". See page no. 203-205.