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Not that I have ever seen this used anywhere (so this might be the wrong direction), but since the second moment method is essentially using Cauchy-Schwarz on the variable $X=X 1_{X>0}$, can't we use Holder's inequality on this expression to obtain higher order inequalities? edit: and obtain something like $P(X>0)\geq (\frac{E[X]^p}{E[X^p]})^{1/(p-1)}$ (assuming $X\geq 0$ and $p>1$).