$\max\{X_1,X_2,\cdots ,X_n\}\xrightarrow{a.s.}x^*_F$ [closed]

Question

I have the following question at hand:

Let $\{X_n\}_{n\ge 1}$ be an iid sequence of random variables with common distribution function $F$ ; such that $x^*_F:=\sup\{x\ : F(x)<1\}<\infty.$ Show that $\max\{X_1,X_2,\cdots ,X_n\}$ converges almost surely to $x^*_F.$

How to attack this problem?

Answer 1

The maximum $M_n:=\max(X_1,\cdots,X_n)$ has distribution $F^n(x)$. Fix $\epsilon>0$, so: \begin{align*} P(|M_n-x_F^*|>\epsilon)&=P(M_n-x_F^*>\epsilon)+P(M_n-x_F< -\epsilon)\\ &\leq 1-F^n(x_F^*+\epsilon)+F^n(x_F^*-\epsilon)\\ &=1-1+z^n, \end{align*}

where $z<1$. Thus:

$$\sum_nP(|M_n-x_F^*|>\epsilon)\leq \sum_n z^n=\frac{1}{1-z}<\infty.$$

Now use the fact that when the Borel Cantelli Lemma holds for all $\epsilon>0$, it follows that $M_n\rightarrow x_F^*$ almost surely.