Almost sure convergence of order random variables

Question

OK, let $X_{1},...X_{n}$ be random variable independts and distributed in the same way. Let $m=inf X_{1}$ and $M=supX_{1}$, that means $\forall a>m, {\cal P}(X_{1}<=a)>0$ and $\forall b<M, {\cal P}(X_{1}>=B)>0$. Prove that $X_{(1)}\rightarrow m$ almost sure and $X_{(n)}\rightarrow M$ almost sure Wherre $X_{(1)},...X_{(n)}$ are the random variable $X_{1},..X_{n}$ but in order.

Answer 1

Let $E_n$ be the event that all samples up to $X_n$ are greater than some $a\gt m$. Then

$$ \sum_{n=0}^\infty\mathcal P(E_n)=\sum_{n=0}^\infty\mathcal P(X_1\gt a)^n=\frac1{1-\mathcal P(X_1\gt a)}=\frac1{\mathcal P(X_1\le a)}\;, $$

a finite value, since $\mathcal P(X_1\le a)\gt0$. Thus, by the Borel–Cantelli lemma, the probability for infinitely many of these events to occur is $0$. Thus, almost surely the first order statistic is greater than $a$ at only finitely many $n$. Since this is true for all $a\gt m$, the first order statistic almost surely converges to $m$.