Asymptotic distribution of $\chi^2_{(1)}$

Question

Let $z_1,\dots,z_n$ be i.i.d. $\chi^2$ with $k$ degrees of freedom (Neither $n$ or $k$ are fixed). Is anything known about the distribution of $\min_i(z_i)$ asymptotically, when say $n/k\to C\in(0,\infty)$? For the max, we can relate this to the Gumbel distribution in this asymptotic setting, but the minimum seems more elusive.

Edit: As I noted, there are problems in the intermediate range precisely because we cannot easily exploit the central limit theorem. See for instance this paper on the max.

Answer 1

If you want the limiting distribution of the minimum of a collection of non-identically distributed chi-squares (each with a different degrees-of-freedom parameter), then this thread, Order statistics (e.g., minimum) of infinite collection of chi-square variates? I believe answers exactly the question.

If you want to see what happens to the distribution of the minimum of $n$ identically distributed chi-squares, as both the sample size and their common degrees-of-freedom parameter goes to infinity, then I have to offer the following remarks:

Assume then that $n/k \rightarrow 0$. It is well known that as $k$ increases the standardized chi-square approaches a standard normal distribution. Since in this case $k$ "goes to infinity faster than $n$", "at" infinity its standardized version is a standard normal distribution and its minimum should follow what the minimum of a normal distribution follows: using David & Nagarajah's notation, the CDF is

$$G^*_3(x) = 1-\exp{\{-e^x\}},\;\; -\infty < x < \infty$$

with $X$ being the minimum standardized by some $a_n, b_n$.

Note that $G^*_3(x) = 1 - G_3(-x)$ with $G_3()$ being the CDF of standard Gumbel distribution.

Now assume that $n/k \rightarrow \infty$. Here, $n$ escapes faster than $k$ and "at" infinity, the variable is -well, it is again a normal, because the normal approximation sets in much earlier than infinity. So it appears that we will again obtain the same result as in the $n/k \rightarrow 0$ case.