How to apply law of large numbers for this problem Let $X_1,...,X_n,...$ be independent variable satisfying $P(X_i=0)=P(X_i=1)=\frac{1}{2}$ for all i then denote $Z_i=X_iX_{i+1}$ for all i .I want to show that $\lim_{n\to \infty}\frac{Z_1+Z_2+...+Z_n}{n}=\frac{1}{4}$ a.s by using law of large number. I'm new to this area and struck immediately since $\{Z_i\}$is not independent.
Perhaps Borel-Cantelli lemma can be used here?

really thanks for your help
Hint: $Z_n$'s are not independent but

what about $Z_{2n}$'s or $Z_{2n+1}'s$?

I am not quite sure the answer is 1/4 considering $Z_n$'s are not independent. I think it is 1/2

Got the idea from here, w/c involves Borel-Cantelli Lemmas, but I don't think you need those.
Hint: $Z_n$'s are not independent but

what about $Z_{2n}$'s or $Z_{2n+1}'s$?

I am not quite sure the answer is 1/4 considering $Z_n$'s are not independent. I think it is 1/2

Got the idea from here, w/c involves Borel-Cantelli Lemmas, but I don't think you need those.