Infinite sequence of coin tosses, $\Bbb{P}(\limsup_{n\to \infty} A_n) = 1$

Question

Let $(\Omega, \mathcal{A}, \Bbb{P}) = \otimes_{j=1}^{\infty} ( \{0,1\}, \mathcal{P}(\{0,1\}), \Bbb{P}_j)$ with $\Bbb{P}_j(1)=1/2=\Bbb{P}_j(0)$, i.e. the model for an independent infinite sequence of coin tosses. Fix $r\in \Bbb{N}$ and $(a_1,…,a_r)\in \{0,1\}^r$, and let $A_k = \{ \omega=(\omega_1,\omega_2,…)\in \Omega : \omega_{k+l-1} = a_l, 1\leq l \leq r \}$.

Then $\Bbb{P}(\limsup_{n\to \infty} A_n) = 1$.

My attempt: The set $\limsup_{n\to \infty} A_n$ consists of those $\omega = (\omega_1, \omega_2,…)$ that are contained in infinitely many $A_k$’s, i.e. they contain the sequence $(a_1,…,a_r)$ infinitely often. At first glance, it looks like we might be able to apply Borel-Cantelli, but the $A_k$’s are not independent, so we cannot apply the Lemma. The statement itself seems intuitively clear, but I haven’t been able to formalize it. Since $( \limsup_{n\to \infty} A_n )^c= \liminf_{n\to \infty} A_n^c$, it would also suffice to show that $\Bbb{P}( \liminf_{n\to \infty} A_n^c) = 0$.

I appreciate any help.

Answer 1

Hints:

1. Show that the sets $A_r, A_{2r},\ldots$ are independent.
2. Deduce from $$\sum_{k \in \mathbb{N}} \mathbb{P}(A_{k \cdot r}) = \infty$$ and the Borel-Cantelli lemma that $\mathbb{P}\left( \limsup_{k \to \infty} A_{k \cdot r} \right) = 1$.
3. Conclude that $$\mathbb{P} \left( \limsup_{n \to \infty} A_n \right)=1.$$