CLT for renewal process of independent non identically distributed random variables

Question

Let $A_{k}$ be independent events with probability $P(A_{k})=k^{-\alpha}$ for some $\alpha \in (0,1)$. We define the renewal process as usual: $$T_{0}=0,$$ $$T_{n+1}=\inf (k>T_{n}|A_{k} occur).$$ Also we define $\lambda_{n}$ to be $-1$ if $T_{n}$ is odd and $1$ if it is even. Show that the following partial sum is satisfying CLT under normalization $$S_{n}=\sum_{i=1}^{n}\lambda_{i}\left(T_{i}-T_{i-1}\right).$$ I can show that $\lambda_{n}$ are mixing very well so I think it is sufficient to show a limit when we switch them with an independent series which distributes equally on {-1,1}. Any help or reference for such non identically distributed renewal process would be much appreciated.