A sort of continuation of [Comparing distributions with moments](https://mathoverflow.net/questions/96165/comparing-distributions-with-moments)


Suppose I have some estimates of the moments of a non-negative random variable $X$: $$\log \mathbb{E}(X^n) = n \log n + (\beta-1)n + O(\log n),$$ with $\beta > 0$. Can I conclude that the tail $$\mathbb{P}(X>x) \sim A e^{-c x}$$ for some $A$ and $c$?



Bounds on tails with moments
You need a weaker conjecture, since the cdf $(1+x)e^{-x}$ for $x\ge 0$ has moments of the desired form yet it isn't like $1-Ae^{-cx}$. More generally, the distribution with cdf $(1+x^k)e^{-x}$ has moments with logarithm
$$n \ln n - n + (1/2+k)\ln n + O(1) .$$ Try cdf $\exp(-x+x^{1/2})$.


