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| In [[mathematics]], the '''Hausdorff moment problem''', named after [[Felix Hausdorff]], asks for necessary and sufficient conditions that a given sequence { ''m''<sub>''n''</sub> : ''n'' = 0, 1, 2, ... }
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| be the sequence of [[moment (mathematics)|moments]]
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| :<math>m_n = \int_0^1 x^n\,d\mu(x)\,</math>
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| of some [[Borel measure]] ''μ'' [[support (measure theory)|supported]] on the closed unit interval [0, 1]. In the case ''m''<sub>0</sub> = 1, this is equivalent to the existence of a [[random variable]] ''X'' supported on [0, 1], such that '''E''' ''X''<sup>n</sup> = ''m''<sub>''n''</sub>.
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| The essential difference between this and other well-known moment problems is that this is on a bounded interval, whereas in the [[Stieltjes moment problem]] one considers a half-line [0, ∞), and in the [[Hamburger moment problem]] one considers the whole line (−∞, ∞).
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| In 1921, Hausdorff showed that { ''m''<sub>''n''</sub> : ''n'' = 0, 1, 2, ... } is such a moment sequence if and only if the sequence is '''completely monotonic''', i.e., its difference sequences satisfy the equation
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| :<math>(-1)^k(\Delta^k m)_n \geq 0</math>
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| for all ''n'',''k'' ≥ 0. Here, Δ is the [[difference operator]] given by
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| :<math>(\Delta m)_n = m_{n+1} - m_n.</math>
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| The necessity of this condition is easily seen by the identity
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| :<math>(-1)^k(\Delta^k m)_n = \int_0^1 x^n (1-x)^k d\mu(x),</math> | |
| which is ''≥ 0'', being the integral of an almost sure non-negative function.
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| For example, it is necessary to have
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| :<math>\Delta^4 m_6 = m_6 - 4m_7 + 6m_8 - 4m_9 + m_{10} = \int x^6 (1-x)^4 d\mu(x) \geq 0.</math> | |
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| ==See also==
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| * [[Total monotonicity]]
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| ==References==
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| * Hausdorff, F. "Summationsmethoden und Momentfolgen. I." ''Mathematische Zeitschrift'' 9, 74-109, 1921.
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| * Hausdorff, F. "Summationsmethoden und Momentfolgen. II." ''Mathematische Zeitschrift'' 9, 280-299, 1921.
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| * Feller, W. "An Introduction to Probability Theory and Its Applications", volume II, John Wiley & Sons, 1971.
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| * [[James Alexander Shohat|Shohat, J.A]].; [[Jacob Tamarkin|Tamarkin, J. D.]] ''The Problem of Moments'', American mathematical society, New York, 1943.
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| ==External links==
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| * [http://mathworld.wolfram.com/MomentProblem.html Moment Problem, on Mathworld]
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| [[Category:Probability theory]]
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| [[Category:Theory of probability distributions]]
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| [[Category:Mathematical problems]]
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Hello. Allow me introduce the writer. Her name is Emilia Shroyer but it's not the most female title out there. Doing ceramics is what her family and her appreciate. North Dakota is our beginning place. Hiring is his occupation.
Here is my webpage :: vine.ac