The main goal of this article is to put some known results in a common
perspective and to simplify their proofs.
We start with a simple proof of a result of Vereshchagin saying that
limsupnC(x∣n) equals C0′(x). Then we use the same argument to prove
similar results for prefix complexity, a priori probability on binary tree, to
prove Conidis' theorem about limits of effectively open sets, and also to
improve the results of Muchnik about limit frequencies. As a by-product, we get
a criterion of 2-randomness proved by Miller: a sequence X is 2-random if and
only if there exists c such that any prefix x of X is a prefix of some
string y such that C(y)≥∣y∣−c. (In the 1960ies this property was
suggested in Kolmogorov as one of possible randomness definitions.) We also get
another 2-randomness criterion by Miller and Nies: X is 2-random if and only
if C(x)≥∣x∣−c for some c and infinitely many prefixes x of X.
This is a modified version of our old paper that contained a weaker (and
cumbersome) version of Conidis' result, and the proof used low basis theorem
(in quite a strange way). The full version was formulated there as a
conjecture. This conjecture was later proved by Conidis. Bruno Bauwens
(personal communication) noted that the proof can be obtained also by a simple
modification of our original argument, and we reproduce Bauwens' argument with
his permission.Comment: See http://arxiv.org/abs/0802.2833 for the old pape