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Suppose we have distributions $p(x)$ and $q(x)$ both supported on integers in $[-n, +n]$. We want $p$ and $q$ to have statistical (total variational) distance of at most $\epsilon$.

Is there a choice of $k$, depending on $\epsilon$ and $n$, such that the distance requirement can be ensured provided that $p$ and $q$ have the same moments up to order $k$ ? How small can $k$ be?

What if, instead of requiring small statistical distance, we wish the weaker guarantee that $p$ and $q$ are $\epsilon$-close in CDF? (i.e., the CDF of the two distributions differ by at most $\epsilon$ at every point).

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Here's an article that is related to your question: How well do moments determine a distribution?

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