We formulate effective versions of the Borel–Cantelli lemmas using a coefficient from Kolmogorov complexity. We then use these effective versions to lift the effective content of the law of large numbers and the law of the iterated logarithm.
"The Borel–Cantelli Lemmas, Probability Laws and Kolmogorov Complexity." Ann. Probab. 29 (4) 1426 - 1434, October 2001. https://doi.org/10.1214/aop/1015345756