Large deviation theory:
Sanov theorem:
large deviation principle and rate function for the "empirical vector"
through the Gartner-Ellis theorem; discrete and continuous random
variables (section 4.2)
Jaynes maximum entropy
principle for independent particle through Sanov theorem for single particle energy distribution
and contraction principle (example 5.1)
Reference review: "The large deviation approach to statistical mechanics" by H. Touchette (Phys. Rep. 2009)