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.
Oldrich Alfonso Vasicek. "A Conditional Law of Large Numbers." Ann. Probab. 8 (1) 142 - 147, February, 1980. https://doi.org/10.1214/aop/1176994830