Let $X_k$ be sequence of independent random variables and suppose $X_k$ has Poisson distribution with mean $k^r$.
For which values of $r$, \begin{equation} \lim_{n\to\infty}\frac{\sum_{k=1}^n X_k}{\sum_{k=1}^n k^r} = 1 \end{equation} with probability one.
If $r<-1$, then the sum is convergent, hence it seems to converge to a random variable instead a constant.
It looks to me as an application of Strong Law of Large Numbers, however, $X_k$'s are not identically distributed. Do I need to scale $X_k$'s?
Brief explanation will help a lot, thank you.