I'm learning strong law of large number recently and there is an exercise that I know how to solve the main part but for a detail I don't know how to prove. The question is:
$y_i,i \geq 1$, are i.i.d r.v.s with $N(0,1)$ distribution. Define $x_1= \frac{y_1+y_2}{2}, x_2= \frac{y_2+y_3}{2},\ldots,x_n= \frac{y_n+y_{n+1}}{2}$. Show that: $\bar x_n \rightarrow c$ as $n \rightarrow \infty$ for some constant $c$ and identify $c$.
I intend to use this large law of number proposition:
$x_i,i \geq 1$, is i.i.d r.v.s with $E(x_i)=\mu_i$ and $\operatorname{Var}(x_i)=\sigma_i^2$ and $\operatorname{corr}(x_i,x_j) \leq \rho_k, k=j-i$. Define further $\bar \mu_n=\frac{\sum\mu_i}{n}$. If $\sum \rho_k < \infty$ and $\sum \frac{\sigma_i^2}{i^2} (\log i)^2 < \infty$, then $\bar x_n-\bar \mu_n$ converges almost surely to $0$ as $n \rightarrow \infty$
I prove that $E(x_i)=0, \operatorname{Var}(x_i)=\frac{1}{2}, \bar\mu_n=0$.
And I try to verify $\sum \rho_k < \infty$ and $\sum \frac{\sigma_i^2}{i^2} (\log i)^2 < \infty$. So that if both hold, $\bar x_n$ will converge in probability to $\bar\mu_n$, which is $0$. So $c$ would be $0$.
For the former, I can prove $\rho_k =1$ as $k=0$ or $1$, and $ \rho_k =0$ as $k>1$. Hence, $\sum \rho_k =2 < \infty$. But I don't know how to prove $\sum \frac{\sigma_i^2}{i^2} (\log i)^2 < \infty$. I know $\sigma_i^2=\frac{1}{2}$, and can be taken out of the summation. But what about $\sum \frac{(\log i)^2} {i^2} < \infty$? Can someone show me how to prove this?
Thank you so much!!!
Hint: $ log \, i=3log \,i^{1/3}<3(i^{1/3}-1)<3i^{1/3}$.