Convergence weakly to exponential random variable for this r.v.

Question

Can I get some insight of how to solve this problem?

Let $X_1, X_2, X_3, ...$ be i.i.d. copies of uniform random variable on $[0, 1]$. Let $M_n = \text{min}_{1\leq i <j \leq n} |X_i - X_j|$. Show $n^2M_n$ converges to $e^{-x}1_{x\geq 0}$ in distribution.

What I have done is:

$$ P(n^2M_n \leq y) = P(M_n \leq \frac{y}{n^2}) = P(\text{min}_{1\leq i < j \leq n}|X_i - X_j| \leq \frac{y}{n^2}),$$ where $$ \begin{align*} P(\text{min}_{1\leq i < j \leq n}|X_i - X_j| \leq \frac{y}{n^2}) = 1 - P(\text{min}_{1\leq i < j \leq n}|X_i - X_j| > \frac{y}{n^2}) \\ \ = 1 - P(\cap_{1 \leq i < j \leq n} \{ |X_i - X_j| > \frac{y}{n^2} \}) \end{align*} $$

But now I don't know how to proceed since many of $|X_i - X_j|$ are dependent (e.g $|X_1 - X_2|$ and $|X_1 - X_3|$) so we cannot easily expand the last term. I will appreciate any insight and help, or just simply a direction to go.

Answer 1

That probability is: $$ \left(1-(n-1)\frac{y}{n^2}\right)^n $$ As take $Y_{ni}$ $i\in\{1,2,\ldots,n\}$ to be uniform on $\left[0,1-(n-1)\frac{y}{n^2}\right]$, and order them, and let $Z_{nk}^*=Y_{nk}^*+(k-1)\frac{y}{n^2}$, then the $Z_{nk}^*$ will have the distribution of $X^*_{1},\ldots,X^*_n$ under the condition that $X_{i+1}^*-X_{i}^*>\frac{y}{n^2}$.

$Y^*_{k}$ denotes an ordered sample.

Answer 2

Let $X_{(1)},\dots,X_{(n)}$ be order statistics of $X_1,\dots,X_n$ and let $E_1,\dots,E_{n+1}$ be i.i.d. $\exp(1)$. Then $$n(X_{(i)}-X_{(i-1)})\sim \frac{nE_{i}}{\sum_{k=1}^{n+1}E_k}\overset{a.s.}\longrightarrow E_i$$ Hence $$\min_{i\le n}n^2(X_{(i)}-X_{(i-1)})\overset{d}\longrightarrow \exp(1)$$ as a minimum of $n$ i.i.d. $\exp(1)$ random variables.