Find CDF of minimum dependent identical distributed random variables

Question

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

1. Node A (e.g., orange point) is uniformly random distributed in the red circle with radius $R$.
2. 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:

1. 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.

2. 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!

Answer 1

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.