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  "body": "\n## Statement\n\n1. Let $\\mu$ be a positive Borel measure on $\\R$ with finite [[moments|01XAHPADBA]] $m_n = \\int x^n \\, d\\mu(x)$ and with $m_{2n} \u003e 0$ for all $n \\ge 1$. If\n$$\\sum_{n=1}^{\\infty} m_{2n}^{-1/(2n)} = \\infty,$$\nthen the [[Hamburger moment problem|TBNCFXTMMW]] for $(m_n)$ is [[determinate|H0GY2J14CA]]: $\\mu$ is the only positive measure on $\\R$ with these moments.\n2. Let $\\mu$ be a positive Borel measure on $[0, \\infty)$ with finite moments $m_n$ and with $m_n \u003e 0$ for all $n \\ge 1$. If\n$$\\sum_{n=1}^{\\infty} m_{n}^{-1/(2n)} = \\infty,$$\nthen the [[Stieltjes moment problem|Y9KH8P9JC0]] for $(m_n)$ is determinate: $\\mu$ is the only positive measure on $[0, \\infty)$ with these moments.\n\n## Remarks\n\nThe proof goes through quasi-analytic functions.\nThe condition puts the Fourier–Stieltjes transform of the difference of two solutions in a quasi-analytic class.\nThat transform has all its derivatives zero at $0$, so it vanishes identically.\nExample: for the standard normal distribution, $m_{2n} = (2n - 1)!! \\le (2n)^n$, so $m_{2n}^{-1/(2n)} \\ge (2n)^{-1/2}$ and the series diverges; the normal law is determined by its moments.\nExample: if $\\mu$ has bounded support in $[-R, R]$, then $m_{2n} \\le m_0 R^{2n}$, so every term is at least $m_0^{-1/(2n)}/R$, and the series diverges.\nThe condition is sufficient but not necessary for determinacy.\nThe hypothesis $m_{2n} \u003e 0$ only excludes the measures concentrated at $0$, which are determinate anyway.\n\n## Explanations\n\n### L1 human\n\nThe moments of a distribution may fail to identify it, but only if they grow very fast.\nCarleman's condition is a precise way of saying that they do not grow too fast.\nWhen it holds, two distributions with the same moments are the same distribution.\n\n### L2 human\n\nIt is the test used in practice to check that a law is determined by its moments, for instance before applying the method of moments to prove convergence in distribution.\nA simpler sufficient condition is that the moment generating function is finite near $0$, and that condition implies Carleman's.\nIn the other direction, Krein's condition gives indeterminacy for densities with very heavy tails, such as the log-normal density.\n",
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