Suppose I have two variables $X$ and $Y$ which have continuous p.d.f.s $f$ and $g$ on the positive real line. I know that the moments $\mathrm{E}[X^n] > \mathrm{E}[Y^n]$ for sufficiently large $n$ (and all moments exist). Can I conclude that there exists a $\xi$ s.t. $f(x) > g(x)$ for all $x > \xi$?



Comparing distributions with moments
Certainly not. Let g be any probability distribution function satisfying g(0)=0 and let
f(x)=g(x-1) if x>1 and f(x)=0 if x<1. Then
$$\int\_0^\infty f(x)x^n\,dx=\int\_0^\infty g(x)(x+1)^n\,dx>\int\_0^\infty x^ng(x)\,dx$$
for any n. But it is not necessarily true that g(x-1)>g(x) for large x.


