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Take nonnegative random variables $X$ whose first $K$ moments have bounds:

$\mu^k\leq E[X^k]\leq c\mu^k$ for each $k=1,\dots,K$.

In this case what is an upper bound for $P(X\leq O(\mu))$?


I am aware of a paper[1] that states the result that the least upper bound is given by the coefficients, but the formula given is very complicated and depends on the exact moments.

[1]: Bounds on a Distribution Function when its First n Moments are Given

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  • $\begingroup$ Example: For $K=4$ and a lognormal distribution $X$ whose log is $N(1,1/2)$, (wolframalpha.com/input/…), we can take $\mu=e^{9/8}$, $c=e^{3/2}$, and we're looking for bounds on probabilities like $P[X\le\mu]=P[N(1,1/2)\le 9/8] = \Phi(1/4)$ in terms of $\mu$ and $c$. $\endgroup$
    – user44143
    Aug 22, 2019 at 18:44
  • $\begingroup$ What is the connection to information theory? And what is the context which makes it natural to use $\mu^k$ as a lower bound, without any coefficient on that term? The equation in the post forces $c \ge 1$, in which case it is trivial that "the least upper bound is given by the coefficients", in the sense that any probability is less than $c$. $\endgroup$
    – user44143
    Aug 22, 2019 at 19:02

1 Answer 1

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Have a look at:

Moment information for probability distributions, without solving the moment problem, II: Main-mass, tails and shape approximation

P.N.Gavriliadis et al., Journal of Computational and Applied Mathematics 229(1), 2009

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