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Let $\mu_n = \langle x^n \rangle$ and $\kappa_n$ be the moments and cumulants of a probability density function $p(x)$.

Given a finite number of the moments, $\mu_1, \dots, \mu_N$, or equivalently a finite number of the cumulants $\kappa_1, \dots, \kappa_N$, I want to approximate the probability density function $p(x)$. The approximation should improve as $N$ increases.

Is there an approximation scheme that allows me to reconstruct $p(x)$ from a finite number of moments with this property?

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  • I am looking for something similar. In my case it doesn't have to be moments or cumulants. I will happy to have any series expansion that improves with more terms, but still generates valid distributions without an infinite expansion. Have you found something. – Cagdas Ozgenc Jan 06 '17 at 14:50

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