The class of concentration of measure inequalities is a fundamental tool in modern probability (and any field that uses probability, e.g., random matrix theory, theoretical computer science, statistics, high-dimensional geometry, combinatorics, etc.). As explained in this blog post of Scott Aaronson, these are basic ways in which one "upper bounds the probability of something bad", and often the bounds are exponential or even gaussian in nature when one is far away from the mean (or median) and there are many independent (or somewhat independent) variables involved. Examples of such inequalities include
- The Chernoff inequality and its relatives (Hoeffding, Bernstein, Bennett, etc.)
- Azuma's inequality
- MacDiarmid's inequality
- Levy's inequality
- Talagrand's concentration inequality
A standard reference in the subject is
Ledoux, Michel, The concentration of measure phenomenon, Mathematical Surveys and Monographs. 89. Providence, RI: American Mathematical Society (AMS). x, 181 p. (2001). ZBL0995.60002.