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.

P. Embrechts, C. Kluppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer Verlag, New York, 2004.

Enrique Castillo, Ali S. Hadi, N. Balakrishnan, Jose M. Sarabia, Extreme Value and Related Models with Applications in Engineering and Science. John Wiley and Sons, 2004


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