# Questions tagged [moment-problem]

The moment problem arises as the result of trying to invert the mapping that takes a measure $μ$ to the sequences of moments, and to resolve the problem of determinacy of such measure.

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### Moment based characterization of invariance

$X$ is compact, any measure $\mu$ is uniquely determined by its moments \begin{align} y_{\alpha}=\int x^{\alpha} d\mu \end{align} where $x^{\alpha}=x_1^{\alpha_1}x_2^{\alpha_2}\dots x_n^{\alpha}$ is ...
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### Convergence of moments implies convergence of the measure in the space of tempered distributions?

Let $\mathcal{S}(\mathbb{R}^d)$ be the Schwartz space and $\mathcal{S}'(\mathbb{R}^d)$ be the space of tempered distributions. Denote by $\{\mu_k\}_{k=1}^\infty$ a sequence of Borel probability ...
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### Fourier transform of indeterminate measure

Consider the following probability measure $d\mu(x) = \frac{1}{\sqrt{2\pi}}x^{-1}\exp\Bigl(-(\ln{x})^2/2\Bigl)\chi_{(0,+\infty)}(x)$ whose moments are given by $\mu_n = e^{n^2/2}$. As discovered by ...
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### Some questions about the number of moments for random variables

I'm reading a paper about econometrics and meeting some questions, and I'm not sure I can describe them perfect. $s_t$ is a scaled score function, and $f_t$ is a stochastic time-varying parameter, and ...
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### Find the moment generating function of a non-canonized gaussian at a specific point t=0

I’ve been trying to solve this problem and hit an integral I’m struggling with. Is my approach the right one? how would you continue? The Question: Let $X \sim N_{[1,4]}$ gaussian random variable. ...
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### Determining function from expectation of its powers

I have a problem concerning reconstructing a function from the expectation of its powers. The scalar version of this problem is as follows: Given a probability measure $\mu(x)$ over $\mathbb{R}^n$ ...
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### Can we retrieve $X$ from its odd moments?

The moment problem asks whether there exists a random variable $X$ with given moments. One way to do this: if $X$'s MGF converges about a neighborhood of $0$, we know the MGF uniquely characterizes $X$...
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### draw random vector for spherically symmetric exponential distribution

My goal is to draw a random (3-dimensional) vector $X$ from a spherically symmetric exponential distribution $$X \sim f_X(x) = \frac{\lambda^3}{8\pi}e^{-\lambda |x|}$$ with $x\in\mathbb{R}^3$. My ...
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### Differences of moment functionals

For which linear functionals $L:\mathbb{R}[x_1,\ldots,x_n]\to\mathbb{R}$ can we find Radon measures $\mu_1$ and $\mu_2$ such that $L(f)=\int f d\mu_1-\int f d\mu_2$ (both subtrahend and minuend should ...
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My question is, given a real random variable $u$, then can we find a discrete random variable $\xi$ with finite possible values, such that $u$ and $\xi$ have the same first $2k$ moments? According to ...
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### Conditional second moment normal distribution

Can someone help me find the expression of $$E(Z^2 | Z \leq 0) \quad when \quad Z\sim N(0,1)$$ I'm aware that $$E(Z | Z \leq 0) = \frac{2}{\sqrt{2\pi}} \quad , \,\, Z \sim N(0,1)$$ using the ...
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