The close relationship between H(p) and E_p(K(x)).
THE COMPLEXITY OF FINITE OBJECTS AND THE DEVELOPMENT OF THE CONCEPTS OF INFORMATION AND RANDOMNESS BY MEANS OF THE THEORY OF ALGORITHMS
a classic 1970 paper by Zvonkin and Levin on Kolmogorov complexity. Proposition 5.1 gives the result that in the limit K(x1...xn)/n converges to H(p) where x's are drawn independently from p.
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