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  "body": "\n## Problem\n\nLet $(m_n)_{n \\ge 0}$ be a [[sequence|0G9EKM0E15]] of real numbers.\nDecide whether there is a positive Borel [[measure|PR11BZQQPG]] $\\mu$ on the real line $\\R$ such that $x^n$ is $\\mu$-integrable and\n$$m_n = \\int_{\\R} x^n \\, d\\mu(x) \\quad \\text{for all } n \\ge 0.$$\nThe numbers $m_n$ are then the [[moments|01XAHPADBA]] of $\\mu$.\n\n## Remarks\n\nThe existence question is solved.\nHamburger proved that such a measure exists if and only if, for every $n \\ge 0$, the Hankel matrix $(m_{j+k})_{j,k=0}^{n}$ is [[positive semidefinite|01MMVW8CKC]].\nEquivalently, $\\sum_{j,k=0}^{n} m_{j+k} c_j c_k \\ge 0$ for all $n$ and all real numbers $c_0, \\ldots, c_n$, since this sum is $\\int \\left( \\sum_j c_j x^j \\right)^2 d\\mu$ when $\\mu$ exists.\nUnlike the Hausdorff moment problem on $[0, 1]$, the measure need not be unique.\nThe moment problem is called determinate when it is unique.\nCarleman's condition $\\sum_{n \\ge 1} m_{2n}^{-1/(2n)} = \\infty$ is sufficient for uniqueness.\nThe moments of the standard normal distribution satisfy it, so that distribution is determined by its moments.\nThe log-normal distribution is a classical example whose moments do not determine it.\n\n## Explanations\n\n### L1 human\n\nThe problem asks which lists of numbers are the moments of some mass distribution spread over the whole real line.\nThe answer is a positivity test: certain square tables built from the list must behave like squared lengths, never giving a negative value.\nOn an unbounded line, two different distributions can share all their moments.\nSo the moments may fail to pin down the distribution, which matters in probability when one tries to identify a law from its moments.\n",
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      "first_stated": "1920",
      "posed_by": "Hans Hamburger"
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    "kind": "problem",
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      "44A60",
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    ],
    "name": "Hamburger moment problem",
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      "wikipedia": "Hamburger moment problem"
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        "url": "https://en.wikipedia.org/wiki/Hamburger_moment_problem"
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  "msc": [
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  "name": "Hamburger moment problem",
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