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I would like to know if there exists a measure $\rho$ on the positive real line such that its moments $\int_0^{\infty} x^j d\rho(x)$ are equal to a constant (for example equal to one) for all $j=0,\dots,n,\dots$ (or for $j\leq n$ for any $n$). In other words, if there exists a density function (equal to zero on the negative real part) such that its characteristic function is $e^{it}$.

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    $e^{it}$ is a continuous map, and positive define, which is $1$ at $0$, so it's a characteristic function. Now, we have to see whether the law whose characteristic function is this map is concentrated on the positive real line. Well, $\rho=\delta_1$ works, but it seems too trivial.2012-07-19
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    and how to see that? Or any reference for that?2012-07-19
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    The first assertion comes from a theorem due to Bochner. But it's not necessary here.2012-07-19
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    Notice that constant moments imply zero variance. What does it tell you about the purported density function?2012-07-19
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    Got something from an answer below?2012-07-25

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