Proof that distribution has power law tails from having infinite moments Is the fact that the 2nd (or higher) moment of a distribution is infinite (while, say,the first moment is finite) proof that the distribution has power law tails? 
Thank you in advance. 
 A: If $E[|X|^\alpha]$ is finite then $P[|X|\geqslant x]=o(1/x^\alpha)$ when $x\to\infty$. The reverse implication is not true.
If $E[|X|]$ is finite and $E[|X|^\alpha]$ is infinite for some $\alpha\gt1$, then one can say that $\lim\limits_{x\to\infty}xP[|X|\geqslant x]=0$ and that $\limsup\limits_{x\to\infty}x^\beta P[|X|\geqslant x]\ne0$ for some $\beta\gt1$.
A: I see we have here a discussion between a mathematician, detail oriented for exactness of the result and a physicist who has a global vision, but no idea of the zoology of ackward distributions a mathematician is able work out...
Let us state a precise mathemtical question: assume that a distribution has finite moments until a given degree m, then infinite for alpha > m (the case of alpha = m exactly can be finite or infinite). Assume also that the pdf f(x) is decreasing at infinity, in order to avoid oscillations. We also assume that the distribution is one-sided (support in IR+) in order to avoid compensation effects between the two sides. Then is there an exponent m' and a constant k such that f(x) > k x^-m' for any x ?
This would be an interesting mathematical exercise ! My guess is that the answer is still negative, though the general idea is very much valid.
