Moment Method

The first and second moment methods are applications of the Chebyshev Inequality. Consider a non-negative and integer-valued Random Variable sequence $(X_n)$.

First Moment Method

If $\mathbb{E}{X}_n\to 0$, then $P(X_n>0)\to 0$.

Proof: By Markov Inequality, we have

$$P(X_n>0)=P(X_n\ge 1)\le \mathbb{E}{X}_n\to 0.$$

Second Moment Method

If $\mathbb{E}{X}_n\nrightarrow 0$ and $\frac{\mathrm{Var}(X_n)}{(\mathbb{E}{X}_n{)}^2}\to 0$, then $P(X_n>0)\to 1$.

Proof: By Chebyshev Inequality, we have

$$P(X_n\le 0)\le P\left(\left|\frac{X_n-\mathbb{E}{X}_n}{\mathbb{E}{X}_n}\right|\ge 1\right)\le \frac{\mathrm{Var}(X_n)}{(\mathbb{E}{X}_n{)}^2}\to 0.$$

Another statement of the second moment method is

$$P(X>0)\ge \frac{(\mathbb{E}X{)}^2}{\mathbb{E}[X^2]}.$$