# Inequalities that show if a distribution decays slowly

Often, one is often interested in theorems/inequalities of the following kind:

Let $X$ be a random variable then the probability that $X$ is close to typically $\mu$ (or larger than some constant) is small. In other words the theorem shows that $X$ decays rapidly in some sense. A typical example is Chebyshev's inequality, which states essentially that $X = \mathbf E[X] + O(\sqrt{\mathbf{Var}[X]})$.

Chapter 1 in Additive combinatorics by Tao and Vu provides lots of interesting results of this kind. (My own interest comes from combinatorics.) My question concerns theorems of the opposite kind:

What are some good theorems that can be used to show that $X$ decays slowly?

For instance, I would be interested in theorems of the sort $\mathrm{Pr}( |X-\mu|> a) > b$ (or without the absolute value), where $a$ and $b$ depend on the moments of $X$. (I am particularly interested in discrete distributions.)

One extremely simple example is: $$\mathrm{Pr}\big( |X - \mathbf E[X]| > \mathbf{Var}(X)^{1/2} \big) > 0.$$

But surely there must be many others?

The Paley-Zygmund inequality says that if $0 < \lambda < 1$, $X \ge 0$, and $E(X^2) < \infty$, then $$\Pr(X \ge \lambda E(X)) \ge (1-\lambda)^2 \frac{(EX)^2}{E(X^2)} .$$ It is in the book by JP Kahane "Some Random Series of Functions," at the end of Section 6 of Chapter 1.