We have that for a real valued random variable $X$,
$$ P(X=0) \leq \frac{\text{Var}(X)}{\left(\mathbb{E}(X)\right)^2} $$ known as Chebyshev's inequality. Consider a random variable $X \in \{0,1,2,\dots\}$, for example counting subgraphs in a random graph. Sometimes, in this setting, the raw moments $\mathbb{E}(X^{n})$ can grow very quickly, since there is significant dependence between the objects being counted.
I am looking to show that $P(X=0) \to 0$ as $\mathbb{E}(X) \to \infty$ using the above bound, but cannot draw any conclusions, because the above variance grows too quickly compared with the square of the mean. Is there a route one can take to try to use more information about the random variable to conclude what is happening to $P(X=0)$? Are there any examples of this?
One idea is to use oscillating sums of factorial moments $E_{r}(X)=\mathbb{E}((X)_r)$ (see Random Graphs, B. Bollobas, Corollary 1.12), where for even $t$
$$ P(X=0) \leq \sum_{r=0}^{t}(-1)^{r}\frac{E_{r}(X)}{r!}. $$ This idea can be difficult to work without good bounds on all moments since one needs to upper bound even moments, and lower bound odd moments, to upper bound the oscillating sum above. Significant work may be required attempting to find a sufficient lower bound on $E_{r}(X)$ given an upper bound, without the bound going to infinity with $\mathbb{E}(X)$.
