Exclude Some samples for calculating CDF I am calculating the asymptotic cumulative distribution of $M_n = \max(X_1,X_2,\dots,X_N)$. My problem is $X_1,X_2,\dots X_p$ and $X_k,X_{k+1},\dots,X_N$ have non identical CDF for $p<<k$ and $p<<N$. But random variables (RVs) in the middle have identical distribution. So for a large $N=20000,p=128,k=19872$, if I take maximum among  $\widetilde{M}_n = \max(X_{p+1},X_{p+2},\dots,X_{k-1})$, I could use Gumbel distribution. Is there any theorem or paper that justify my action ? Or if I want to exclude some samples, what are the conditions to be fulfilled ? 
Edit: $X_1,X_2,\dots X_p$ has Gamma distribution $\Gamma(k_i,\theta_i)$
  $X_k,X_{k+1},\dots,X_N$ has Gamma distribution  $\Gamma(k_i,\theta_i)$

  All other has Gamma distribution  $\Gamma(k,\theta)$

 A: For convenience, I will denote by $L$ the initial subset of non-identically distributed variables, by $C$ the middle "core" subset of identically distributed ones, and by $R$ the right subset. I also understand that the $1,...,N$ index is just labeling, and not ordering.
First, you have to deal with the issues @whuber raised in the comments: to invoke asymptotic theory, you must contemplate what happens as your sample size increases: what kind of random variables will be added to your sample? Of the $L$-variety, of the $C$-variety, or of the $R$-variety? Or, say, of all varieties in fixed, or random, or unknown, proportions?
Ideally, you would want that 
$$\widetilde{M}_n = M_n \Rightarrow \max(X_{p+1},X_{p+2},\dots,X_{k-1})  = \max(X_1,X_2,\dots,X_N) \tag{1}$$
For this to hold deterministically you would need
$$\max(X_{1},\dots,X_{p},X_{k}\dots,X_{N}) <\max(X_{p+1},X_{p+2},\dots,X_{k-1}) \tag{2}$$
which you cannot deterministically guarantee, since all your variables are Gammas with support unbounded from above.  
Nevertheless, it appears that informally, you sense that since the number of $C$-variables is much-much larger than the number of $L$- and $R$-variables, it may be an acceptable approximation to ignore the latter two groups. This is not necessarily so. It has to do with the previous issue of how your sample is going to be populated as it increases in size, and also, it has to do with the parameters of the Gammas involved: the smaller the values of the shape and scale parameters of a Gamma distribution, the more concentrated is the distribution to the left, near zero and small values. So if the common parameters $(k,\,\theta)$ are "too small", while some of the $(k_i,\,\theta_i)$ are "too large", then, even if the $L$- and $R$- variables are much fewer in number, they generate larger values with much higher probability than the $C$-variables.  
On the other hand, if it can be assumed that $(k,\,\theta)$ are sufficiently larger than the various individual parameters $(k_i,\,\theta_i)$, (especially $\theta$ which enters the variance squared), then intuitively inequality $(2)$ starts to have "large probability" of being true, perhaps sufficiently large to make the use of $\widetilde{M}_n$ acceptable (in the sense of not being misleading)... if more over, the issue of the sample's asymptotic composition favors the $C$-variables.  
Are there answers to these issues, (or are you willing to make assumptions about them)?
