Order statistics (e.g., minimum) of infinite collection of chi-square variates? This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.).  (And hopefully I can edit later!)  I tried to find references, and tried to solve myself using induction, but failed at both.
I'm trying to simplify a distribution that seems to reduce to an order statistic of a countably infinite set of independent $\chi^2$ random variables with different degrees of freedom; specifically, what is the distribution of the $m$th smallest value among independent $\chi^2_2,\chi^2_4,\chi^2_6,\chi^2_8,\ldots$?
I would be interested in the special case $m=1$: what is the distribution of the minimum of (independent) $\chi^2_2,\chi^2_4,\chi^2_6,\ldots$?
For the case of the minimum, I was able to write the cumulative distribution function (CDF) as an infinite product, but can't simplify it further.  I used the fact that the CDF of $\chi^2_{2m}$ is $$F_{2m}(x)=\gamma(m,x/2)/\Gamma(m)=\gamma(m,x/2)/(m-1)!=1-e^{-x/2}\sum_{k=0}^{m-1}x^k/(2^k k!).$$  (With $m=1$, this confirms the second comment below about equivalence with an exponential distribution with expectation 2.)  The CDF of the minimum can then be written as $$F_{min}(x) = 1-(1-F_2(x))(1-F_4(x))\ldots = 1-\prod_{m=1}^\infty (1-F_{2m}(x)) $$ $$= 1- \prod_{m=1}^\infty \left(e^{-x/2}\sum_{k=0}^{m-1}\frac{x^k}{2^k k!}\right).$$  The first term in the product is just $e^{-x/2}$, and the "last" term is $e^{-x/2}\sum_{k=0}^\infty x^k/(2^k k!)=1$.  But I don't know how (if possible?) to simplify it from there.  Or maybe a totally different approach is better.
Another potentially helpful reminder: $\chi^2_2$ is the same as an exponential distribution with expectation 2, and $\chi^2_4$ is the sum of two such exponentials, etc.
If anyone is curious, I am trying to simplify Theorem 1 in this paper for the case of regression on a constant ($x_i=1$ for all $i$).  (I have $\chi^2$ instead of $\Gamma$ distributions since I have multiplied by $2\kappa$.)
 A: The zeros of the infinite product will be the union of the zeros of the terms.  Computing out to the 20th term shows the general pattern:

This plot of the zeros in the complex plane distinguishes the contributions of the individual terms in the product by means of different symbols: at each step, the apparent curves are extended further and a new curve is started even further left.
The complexity of this picture demonstrates there exists no closed-form solution in terms of well-known functions of higher analysis (such as gammas, thetas, hypergeometric functions, etc, as well as the elementary functions, as surveyed in a classic text like Whittaker & Watson).
Thus, the problem might be more fruitfully posed a little differently: what do you need to know about the distributions of the order statistics?  Estimates of their characteristic functions?  Low order moments?  Approximations to quantiles?  Something else?
A: 
what is the distribution of the minimum of (independent) $\chi^2_2,\chi^2_4,\chi^2_6,\ldots$?

Apologies for arriving some 6 years late. Even though the OP has likely now moved onto other problems, the question remains fresh, and I thought I might suggest a different approach.

We are given $(X_1, X_2, X_3, \dots)$ where $X_i \sim \text{Chisquared}(v_i)$ where $v_i= 2i$ with pdf's $f_i(x_i)$:

Here is a plot of the corresponding pdf's $f_i(x_i)$, as the sample size increases, for $i = 1 \text{ to } 8$:

We are interested in the distribution of $\text{min}(X_1, X_2, X_3, \dots)$.
Each time we add an extra term, the pdf of the marginal last term added shifts out further and further to the right, so that the effect of adding more and more terms becomes not only less and less relevant, but after a just a few terms, becomes almost negligible -- on the sample minimum. This means, in effect, that only a very small number of terms are likely to actually matter ... and adding additional terms (or the presence of an infinite number of terms) is largely irrelevant for the sample minimum problem.
Test
To test this, I have calculated the pdf of $\text{min}(X_1, X_2, X_3, \dots)$ to 1 term, 2 terms, 3 terms, 4 terms,  5 terms,  6 terms, 7 terms, 8 terms, to 9 terms,  and to 10 terms. To do this, I have used the OrderStatNonIdentical function from mathStatica, instructing it here to calculate the pdf of the sample minimum (the $1^{\text{st}}$ order statistic) in a sample of size $j$, and where parameter $i$ (instead of being fixed) is $v_i$:


It gets a bit complicated as the number of terms increase ... but I have shown the output for 1 term (1st row), 2 terms (second row), 3 terms (3rd row) and 4 terms above.
The following diagram compares the pdf of the sample minimum with 1 term (blue), 2 terms (orange), 3 terms, and 10 terms (red). Note how similar the results are with just 3 terms vs 10 terms:

The following diagram compares 5 terms (blue) and 10 terms (orange) -- the plots are so similar, they obliterate each other, and one cannot even see the difference:
 
In other words, increasing the number of terms from 5 to 10 has almost no discernible visual impact on the distribution of the sample minimum.
Half-Logistic Approximation
Finally, an excellent simple approximation of the pdf of the sample min is the half-Logistic distribution with pdf: 
$$g(x) = \frac{2 e^{-x}}{\left(e^{-x}+1\right)^2} \quad \text{ for } x>0$$ 
The following diagram compares the exact solution with 10 terms (which is indistinguishable from 5 terms or 20 terms) and the half-Logistic approximation (dashed):

Increasing to 20 terms makes no discernible difference.
