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

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.

• the condition $n/k \to C$ looks erroneous and redundant. – Zhanxiong Feb 3 '15 at 0:29
• but here you stated $k$ as the degrees of freedom of $\chi^2$ random variable, should be a fixed integer. And since you are interested in the limiting distribution of $Z_{(1)}$, it is automatically the case $1/n \to 0$. – Zhanxiong Feb 3 '15 at 1:29
• I did not say that $k$ was fixed. This distribution is relevant when considering the diagonals of covariance matrices, for instance, in which context one often lets $k$ and $n$ scale to $\infty$ together. – Lepidopterist Feb 3 '15 at 1:34
• all right, I see. – Zhanxiong Feb 3 '15 at 1:36

## 1 Answer

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.

• you addressed the two cases that I was not interested in! I am interested in $n/k\to C\in(0,\infty)$. Unless you are suggesting that the "extreme" cases point to the general solution? It is known (see here: wisdom.weizmann.ac.il/~nadler/Publications/…) that there are problems in the intermediate range for the maximum. – Lepidopterist Feb 3 '15 at 18:02
• @Lepidopterist Strange... I remember a comment that is now nowhere to be seen where the two cases were explicitly mentioned and I understood that these were the cases of interest. Tricks memory plays... Anyway my answer does not deal with the intermediate case, and I cannot say whether it applies or not. – Alecos Papadopoulos Feb 3 '15 at 18:41
• I deleted the comment, because it was confusing and I guessed that it could have been the source of your confusion. I was pointing out that people are interested in various regimes, for example two of which you just did. But I was trying to point out that my question asked for a different one. – Lepidopterist Feb 3 '15 at 18:43
• @Lepidopterist Ok. Now that this is cleared, I would appreciate if you edited correspondingly the edit you made in the body of the question, since I did not ignore the actual question, I just got confused by a comment of yours. Finally, to return to the question, can you also clarify which of the two situations are of interest? The one where the sample consists of non-identically distributed variables, or the one where it consists of identically distributed ones? – Alecos Papadopoulos Feb 3 '15 at 18:50
• Alecos, I'm confused. I stated that the variables are i.i.d. (i.e. identically), and I stated the asymptotic regime I was interested in the body of my question. – Lepidopterist Feb 3 '15 at 18:56