Suppose $X$ is a non-negative random variable with bounded image. I was wondering if anybody knew of any results that could answer a question of the following type: Suppose the $n$-th moment satisfies $$f(n)\leq \mathbb{E}[X^n]\leq g(n).$$ With appropriate assumptions on $f(n)$, $g(n)$, in what sense can one approximate the distribution of $X$? Thanks in advance.
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2$\begingroup$ If the moments of $X$ are close to that of another random variable $Y$, it may be possible to show that the distribution of $X$ is also close to $Y$ by taking apart the proof of the Carleman/moment continuity theorem (see e.g. Theorem 4 in the notes terrytao.wordpress.com/2010/01/05/…). What are some examples of $f$ and $g$ for which you'd like to draw conclusions? $\endgroup$– Brad RodgersCommented Jun 14, 2016 at 7:28
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