Assume that the first $n$ moments $m_1,\dots\,m_n$ of a random variable $X\in\mathbb{R}$ are known, but not its probability density function $p(x)$.

Does there exist a methodology to characterize the tail behaviour of the p.d.f ?

For example whether $p(x)\sim |x|^{-\theta}$ or $p(x)\sim e^{-\theta \, |x|^\gamma}$.

  • $\begingroup$ Did you look at truncated moment problems? $\endgroup$ – Davide Giraudo Apr 10 '15 at 20:39

If two distributions $X$ and $Y$ have the same moments from 0 up to $n$, then the CDFs of the two functions differ asymptotically by less than $x^{-n}$. See this page for details and references.

  • 1
    $\begingroup$ That's interesting and useful, but does it tell us anything at all about the tails of either distribution? $\endgroup$ – whuber Apr 10 '15 at 21:30

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