Suppose $F$ is a Pareto distribution with scale parameter $x_m$ and shape parameter $\alpha$. Assume $X_1, X_2 , \dots, X_n$ are iid random variables drawn from $F$.
Let $S_n(k) = X_1 ^k + X_2 ^k + \dots + X_n ^k $.
Can we say anything about $\frac{S_n(k)}{S_n(1)}$ as $ n \rightarrow \infty$ ?
Will it be easier to solve, if the distribution $F$ is power law but bounded (that is, $(\forall i)[1 \leq X_i \leq n]$)?