$X_1,X_2,\dots,X_n$ are i.i.d r.v.$P(X_n=-1)=p_0,P(X_n=2^k-1)=p_k, k\in N. $where $p_0=1-\sum\limits_k p_k,p_k=\frac{1}{2^kk(k+1)}.$ How to prove \begin{equation} \frac{S_n}{\frac{n}{\log_2n}}\xrightarrow{P} -1. \end{equation} where $S_n=X_1+\dots+X_n$
I want to use WLLN to find the similar form and
$EX_n=-p_0+\sum\limits_k\frac{2^k-1}{2^kk(k+1)}=-1+\sum\limits_k\frac{1}{k(k+1)}=0.$
By the Khinchin's law :$\frac{S_n}{n}\xrightarrow{P} 0$. But I don't know how to get its rate.
Thank you for your help.
I find the similar question from Durrett's book, and I just post the answer by Durrett.
This is question.
And theorem 2.2.6
Another similar example in book is:
