It is shown that, when conditional on a set of given average values, the frequency distribution of a series of independent random variables with a common finite distribution converges in probability to the distribution which has the maximum relative entropy for the given mean values.
"A Conditional Law of Large Numbers." Ann. Probab. 8 (1) 142 - 147, February, 1980. https://doi.org/10.1214/aop/1176994830