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An infinite binary sequence has randomness rate at least $\sigma$ if, foralmost every $n$, the Kolmogorov complexity of its prefix of length $n$ is atleast $\sigma n$. It is known that for every rational $\sigma \in (0,1)$, onone hand, there exists sequences with randomness rate $\sigma$ that can not beeffectively transformed into a sequence with randomness rate higher than$\sigma$ and, on the other hand, any two independent sequences with randomnessrate $\sigma$ can be transformed into a sequence with randomness rate higherthan $\sigma$. We show that the latter result holds even if the two inputsequences have linear dependency (which, informally speaking, means that allprefixes of length $n$ of the two sequences have in common a constant fractionof their information). The similar problem is studied for finite strings. It isshown that from any two strings with sufficiently large Kolmogorov complexityand sufficiently small dependence, one can effectively construct a string thatis random even conditioned by any one of the input strings.

Extracting the Kolmogorov Complexity of Strings and Sequences from Sources with Limited Independence