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Suppose $Y=\max\{X_1, X_2,\dots,X_N\}$ where all $X_i$ are independent and follows Erlang distribution. I know that extreme value theory deals with maximum of random variables. Can anybody tell me, hopefully with reference, $Y$ will follow which extreme value distribution (Gumbel, Weibull or Frechet) ?

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If you mean the Erlang distribution with density $$f(x)={\lambda^kx^{k-1}e^{-\lambda x}\over(k-1)!}$$ for parameters $\lambda>0$ and positive integer $k$, then the max of $N$ independent Erlangs has an exponential tail, so the Erlang belongs to the domain of attraction of the Gumbel distribution.

To prove this, see this CrossValidated post, which states a sufficient condition for the max to converge to Gumbel. It appears that the Erlang satisfies this condition.

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We can consider Erlang distribution as a example of PH-distribution (phase type). This way allows to use matrix representation for PH-distribution and closure properties for one. For instance, the maximum $X_{max}$ of two PH random variables $X_1$ and $X_2$ has PH distribution.

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