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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, $(\forall i)[1 \leq X_i \leq n]$)?

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  • $\begingroup$ My intuition is that the ratio will be constant or a non-degenarate distributed random variable. I know that the sum of power law random variables behaves same as the maxima(in asymptotic sense), therefore the sum of the $k^{th}$ power would be even more skewed towards the maxima. Hence, the two terms would be of the same order. $\endgroup$
    – rajatsen91
    Jun 25, 2014 at 14:35

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