how to show that random variable is almost surely finite? Coin toss problem

Question

Consider the Coin toss problem, i.e. let $Z_{i} : \Omega \rightarrow \{0,1\}$ with $$ Z_{i}\left(\omega\right) = \begin{cases} 1 & \text{if }\omega = H\\ 0 & \text{if } \omega = T \end{cases} $$ be the outcome of the $i$-th coin toss with $\Omega = \{H,T\}$. Assume all the $Z_{i}$ are independent and identical distributed with $$ \mathbb{P}(Z_{i} = 1) = \mathbb{P}(Z_{i} = 0) = \frac{1}{2}. $$ Now define $$ R := \min \{k\geq 1 \mid Z_{k} = 1,Z_{k+1} = 0,Z_{k+2} = 1, Z_{k+3} = 0\}. $$ $R$ is the number of tosses to get the pattern HTHT.

My question ist how to show that $R$ is almost surely finite. Any hints?

Answer 1

Hints:

1. Show that $$A_k := \{Z_{4k}=0, Z_{4k+1}=1, Z_{4k+2}=0, Z_{4k+3}=1\}$$ satisfies $\mathbb{P}(A_k) = 1/16$ for each $k \in \mathbb{N}$.
2. Show that the events $A_k$, $k \geq 1$, are independent.
3. It follows from Step 1 that $$\sum_{k \geq 1} \mathbb{P}(A_k) = \infty.$$ Apply the Borel Cantelli lemma (using Step 2) to conclude that $$\mathbb{P}(A_k \, \, \text{infinitely often})=1.$$
4. Conclude that $\mathbb{P}(R<\infty)=1$.