# Almost sure convergence of AR(1) model

I am trying to solve following problem.

Problem. Suppose that $$X_n = \rho X_{n-1} + \epsilon_n$$ with $$|\rho| < 1$$ and $$X_0 = 0$$, where $$\epsilon_n$$ are iid r.v.'s with mean $$0$$ and variance $$1$$. Show that $$\max_{1\le k \le n} |X_k|/\sqrt{n} \to 0~$$ a.s.

My idea is using Borel-Canteli lemma to show the a.s. convergence.

Since $$\max_{1\le k \le n} |X_k|/\sqrt{n} \to 0 ~\text{ a.s. } \iff \forall\epsilon > 0: P(\max_{1\le k \le n} |X_k|/\sqrt{n} > \epsilon ~\text{ i.o.}) = 0,$$ and $$\{\max_{1\le k \le n} |X_k|/\sqrt{n} > \epsilon ~\text{ i.o.}\} = \{ |X_n|/\sqrt{n} > \epsilon ~\text{ i.o.}\},$$
(is it true?) I think that it is enough to show that $$|X_n|/\sqrt{n} \to 0 ~\text{ a.s. }$$

To apply BC lemma, I am trying to bound $$\sum_n P(|X_n|/\sqrt n > \epsilon)$$. But Markov inequality only shows that $$P(|X_n|/\sqrt n > \epsilon) \le \frac{var(X_n)}{n\epsilon^2} = \frac{1-\rho^{2n}}{(1-\rho^2)n\epsilon^2}.$$

But this cannot bound $$\sum_n P(|X_n|/\sqrt n > \epsilon)$$. How can I precede? Shall we need a 4th finite moment for $$X_n$$?

Hints: $$X_n=\rho^{n-1}\epsilon_1+\rho^{n-2} \epsilon_2+...+\epsilon_n$$ by iteration. This shows that $$X_n$$ converges a.s.. [ You can use Kolmogorov's Three Series Theorem, for example]. Note that if a sequence $$(x_n)$$ of real numbers is bounded then $$\max \{|x_1|,|x_2|,...|x_n|\} / \sqrt n \to 0$$.