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.
I have some papers that gives a lower bound on tail probabilities, typically on sums of independent random variables. The first paper is based upon the inequality of Paley and Zygmund. http://faculty.missouri.edu/~stephen/preprints/tail.html http://faculty.missouri.edu/~stephen/preprints/compare.html http://faculty.missouri.edu/~stephen/preprints/weibulls.html http://faculty.missouri.edu/~stephen/preprints/tailproc.html http://faculty.missouri.edu/~stephen/preprints/disttail.html I know a lot of other people have done this type of thing as well - look at the references in the last paper. In particular, I was really impressed by the paper by Latala, (1997) Estimation of moments of sums of independent random variables, Ann. Probab., 25, 1502-1513.