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Joseph Miller [16] and independently Andre Nies, Frank Stephan and SebastiaanTerwijn [18] gave a complexity characterization of 2-random sequences in termsof plain Kolmogorov complexity C: they are sequences that have infinitely manyinitial segments with O(1)-maximal plain complexity (among the strings of thesame length). Later Miller [17] showed that prefix complexity K can also beused in a similar way: a sequence is 2-random if and only if it has infinitelymany initial segments with O(1)-maximal prefix complexity (which is n + K (n)for strings of length n). The known proofs of these results are quite involved;in this paper we provide simple direct proofs for both of them.  In [16] Miller also gave a quantitative version of the first result: the0'-randomness deficiency of a sequence {\omega} equals lim inf [n - C({\omega}1 . . . {\omega}n)] + O(1). (Our simplified proof can also be used toprove this.) We show (and this seems to be a new result) that a similarquantitative result is also true for prefix complexity: 0'-randomnessdeficiency equals lim inf [n + K (n) -- K ({\omega}1 . . . {\omega}n)] + O(1).

Prefix and plain Kolmogorov complexity characterizations of 2-randomness: simple proofs