I am currently thinking about the following problem, which seems to be rather elementary, but I was unable to find or come up with a satisfactory solution.
Assume that there are two sorted sequences of numbers, $0 \leq \lambda_1 \leq ... \leq \lambda_d \leq 1$ and $0 \leq \mu_1 \leq ... \leq \mu_d\leq 1$, and $\sum_i \lambda_i = \sum_i \mu_i = 1$ (you might think about the $\lambda$ and $\mu$ as probabilities). Assume that we know \begin{align} \sum_i \lambda_i^k - \sum_i \mu_i^k \leq \epsilon_k \end{align} for all $k=1,2,...,d$ (clearly, $\epsilon_1 = 0$ due to normalization). Is there any way to bound a distance measure between the two sequences, like the total variation distance, or the Kullback-Leibler divergence, just from knowledge of the $\epsilon_k$?
Any help or hint on existing literature would be greatly appreciated!
