# Question about mean squared error of a sum raised to the kth power

Let $$\begin{equation} Q = \sum_{i=1}^{n} \alpha_i \end{equation}$$ be a sum that we want to estimate. Let us suppose we have an algorithm $$\mathcal{A}$$, outputting an estimate $$\hat Q$$, such that $$\begin{equation} \mathbb{E}[(Q - \hat Q)^{2}] = \beta, \end{equation}$$ where the expectation is taken over the randomness of the algorithm.

Now, consider the quantity $$Q^{k}$$ for an integer $$k$$.

We run $$\mathcal{A}$$ to get $$\hat Q$$ and output $$\hat Q^{k}$$ as an estimate for $$Q^{k}$$. Can we bound the quantity $$\begin{equation} \mathbb{E}[(Q^{k} - \hat Q^{k})^{2}], \end{equation}$$ in terms of $$\beta$$ --- where the expectation is again taken over the randomness of the algorithm?

I think it should be more than $$\beta^{k}$$ but I could not prove it.

For simplicity, take each $$\alpha_i$$ to be between $$0$$ and $$1$$.

• For some distributions, and $k$ large enough,\begin{equation}\mathbb{E}[(Q^{k} - \hat Q^{k})^{2}],\end{equation}is infinite, so there is no hope in finding a general relation. Oct 2, 2021 at 7:39
• What might be an example of such a distribution? Oct 2, 2021 at 14:08
• Any Student t distribution. Any Pareto distribution.
– whuber
Oct 2, 2021 at 14:29