Find CDF of minimum dependent identical distributed random variables I'm a post-graduate researcher in Telecommunications and am currently studying Geogeomatric stochastic's applications.
In the process of building systems, I faced the challenge of finding the minimum probability of multiple dependent random variables. Details of it can be explained as follows. As illustrated in the figure below, I have

*

*Node A (e.g., orange point) is uniformly random distributed in the red circle with radius $R$.

*Node $N$ node B (e.g., blue triangle) is uniformly random distributed in the red circle with radius $R$.

Thus, the predetermined probability density distribution (PDF) of the distance between A (or B$_n$, $n=1,2,..N$) and the origin can be easily formulated according to $R$. Here, I denote such these corresponding PDFs as $f_{d_{\rm OA}}(x)$ and $f_{d_{{\rm OB}_n}}(x)$.
My aim is to determine:

*

*The PDF of the unknown distance between node A and one any node B$_n$, i.e., $f_{d_{{\rm AB}_n}}(x)$. In this case, I have fully derived $f_{d_{{\rm AB}_n}}(x)$, based on the conditional probability approach. Note, $f_{d_{{\rm AB}_n}}(x)$ is the same for all $n=1,2,..N$ as my checked from Matlab simulation results.


*The minimum PDF of the unknown distance between A and B$_n$, denoted by $f_{d_{{\rm AB}_{min}}}(x)$. To do this, I tried to compute $F_{d_{{\rm AB}_{min}}}(y) = \Pr[\min\{d_{{\rm AB}_n}\}<y]$ and $f_{d_{{\rm AB}_{min}}}(y) =\frac{ \partial F_{d_{{\rm AB}_{min}}}(y)}{\partial y}$ with the aim of using the obtained PDF $f_{d_{{\rm AB}_n}}(x)$. However, from the figure below, I observe that $d_{{\rm AB}_n}$ is correlated at A. In other words, all random variables $d_{{\rm AB}_n}$ are dependent. Thus, I cannot extract the probability above according to the independent case as
\begin{align}
F_{d_{{\rm AB}_{min}}}(y) &= \Pr[\min\{d_{{\rm AB}_n}\}<y] = 1 - \Pr[\min\{d_{{\rm AB}_n}\}>y] = \Pr[d_{{\rm AB}_1}>y,...,d_{{\rm AB}_n}>y,...,d_{{\rm AB}_N}>y]\\
&\ne 1-\prod_{n=1}^N(1-F_{d_{{\rm AB}_n}}(x)) = 1-(1-F_{d_{{\rm AB}_n}}(x))^N.
\end{align}
So, in this case, could you please recommend a way to solve the problem of the probability above? Approximation or Asymptotic methods are also ok with me.
Thank you for your enthusiasm!

 A: A) The problem can be scaled to have the red circle unitary
Being either A and the B points distributed uniformly inside the circle, then using polar coordinates
the pdf of any point will be
$$
\Pr \left( {{\rm A}{\rm ,B}} \right)\; \sim \;\frac{r}{{\pi R^2 }}drd\alpha
  = \frac{1}{\pi }\frac{r}{R}d\left( {\frac{r}{R}} \right)d\alpha  = \quad \left| \begin{array}{l}
 \;0 < \frac{r}{R} \le 1 \\  \; - \pi  < \alpha  \le \pi  \\ 
 \end{array} \right.
$$
and in fact
$$
\int\limits_{0 < \frac{r}{R} \le 1} {\int\limits_{ - \pi  < \alpha  \le \pi } {\Pr \left( {{\rm A}{\rm ,B}} \right)} }
  = \frac{1}{\pi }\int\limits_{0 < \frac{r}{R} \le 1} {\int\limits_{ - \pi  < \alpha  \le \pi }
 {\frac{r}{R}d\left( {\frac{r}{R}} \right)d\alpha } }  = \frac{{2\pi }}{\pi }\frac{1}{2} = 1
$$
So in the proceeding let's understand the radii to be standardized by $R$. as above
B) The point A can be rotated onto the $x$ axis
Clearly, wlog, the point A can be taken to lay on the $x$ axis, at a standardized coordinate $\rho$, with pdf
$$
\Pr \left( {\rm A} \right) \sim 2\rho \,d\rho
$$
C) B distances wrt A
The distance of $B_k$ from A, always standardized,  reads
$$
\begin{array}{l}
 d_k  = \left\| {{\rm AB}_k } \right\| = \sqrt {\left( {r_k \cos \alpha _k  - \rho } \right)^2  + r_k ^2 \sin ^2 \alpha _k }
  = \sqrt {\left( {r_k ^2  - 2\rho \cos \alpha _k  + \rho ^2 } \right)}  =  \\ 
  = \sqrt {r_k ^2  + \rho ^2 } \sqrt {\left( {1 - \frac{{2\rho }}{{\sqrt {r_k ^2  + \rho ^2 } }}\cos \alpha _k } \right)}  \\ 
 \end{array}
$$
D) Prob given $\rho$
Now we can level out $\alpha$ to find the probability of $d_k$ given $\rho$, and we have better to express that as CDF.
But the integration bounds on $\alpha$ depend on $d_k$,  besides on $\rho$ as illustrated in the sketch below

So,

*

*for $0 < d_k \le 1- \rho$, $\alpha$ is spanning the whole circle;

*for $ 1- \rho < d_k \le 1+ \rho$, $alpha$ is limited to the circular sector shown.

We will obtain a piecewise CDF and as such we shall insert it into {Extreme value computation](https://en.wikipedia.org/wiki/Extreme_value_theory).
I am not going here into the relevant detailed computation.
Once we have the probability of the minmum $d $ given $\rho$, and the probability of $\rho$ as in B),
then it is a matter to integrate over $\rho$ their product.
