Lets $(X_n)_{n\in\mathbb{N}}$ be an iid sequence of standard normal random variables. Define $$M_n=\max_{1\leq i\leq n} X_i.$$
Prove that $$\lim_{n\rightarrow\infty} \frac{M_n}{\sqrt{2\log n}}=1\quad\text{a.s.}$$
I used the fact that $$\left(\frac{1}{x}-\frac{1}{x^3}\right)e^{-\frac{x^2}{2}}\leq\mathbb P(X_n>x)\leq \frac{1}{x}e^{-\frac{x^2}{2}},$$
and the Borel Cantelli lemmas to prove that
$$\limsup_{n\rightarrow\infty} \frac{X_n}{\sqrt{2\log n}}=1\quad\text{a.s.}$$
I used Davide Giraudo's comment to show $$\limsup_n \frac{M_n}{\sqrt{2\log n}}=1\quad \text{a.s.}$$
I have no idea how to compute the $\liminf$. Borel-Cantelli give us tools to compute the $\limsup$ of sets, I am unsure of how to argue almost sure convergence. Any help would be appreciated.
Best Answer
Your estimate gives that for each positive $\varepsilon$, we have $$\sum_i\mathbb P(M_{2^{i}}>(1+\varepsilon)\sqrt 2\sqrt{i+1})<\infty$$ (we use the fact that $\mathbb P(M_n>x)\leqslant n\mathbb P(X_1\gt x)$).
We thus deduce that $\limsup_iM_{2^{i}}/(\sqrt{2(i+1)})\leqslant 1+\varepsilon$ almost surely. Taking $\varepsilon:=1/k$, we get $\limsup_iM_{2^{i}}/(\sqrt{2(i+1)})\leqslant 1$ almost surely. To conclude, notice that if $2^i\leqslant n\lt 2^{i+1}$, $$\frac{M_n}{\sqrt{2\log n}}\leqslant \frac{M_{2^{i+1}}}{\sqrt{2i}}.$$
For the $\liminf$, define for a fixed positive $\varepsilon$, $$A_n:=\left\{\frac{M_n}{\sqrt{2\log n}}\lt 1-\varepsilon\right\}.$$ A use of the estimate on the tail of the normal distribution show that the series $\sum_n \mathbb P(A_n)$ is convergent, hence by the Borel-Cantelli lemma, we have $\mathbb P\left(\limsup_n A_n\right)=0$. This means that for almost every $\omega$, we can find $N=N(\omega)$ such that if $n\geqslant N(\omega)$, then $$\frac{M_n(\omega)}{\sqrt{2\log n}}\geqslant 1-\varepsilon.$$ This implies that $$\liminf_{n\to \infty}\frac{M_n(\omega)}{\sqrt{2\log n}}\geqslant 1-\varepsilon \quad \mbox{a.e.}$$ Here again, taking $\varepsilon:=1/k$, we obtain that $$\liminf_{n\to \infty}\frac{M_n(\omega)}{\sqrt{2\log n}}\geqslant 1\quad \mbox{a.e.}$$