We correct Miyabe's proof of van Lambalgen's theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van Lambalgen's theorem for Schnorr randomness. It has been claimed in the literature that this corollary (and the analogous result for computable randomness) is a "straightforward modification of the proof of van Lambalgen's theorem." This is not so, and we point out why. We also point out an error in Miyabe's proof of van Lambalgen's theorem for truth-table reducible randomness (which we will call uniformly relative computable randomness). While we do not fix the error, we do prove a weaker version of van Lambalgen's theorem where each half is computably random uniformly relative to the other. We also argue that uniform relativization is the correct relativization for all randomness notions. arXiv:1209.5478v2 [math.LO]

MSC (2010) 68Q30, 03D15, 03D25Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen's Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness (defined in [6] too only by martingales) and truth-table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness.

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We give a unified treatment of the convergence of random series and the rate of convergence of strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk [24]. We consider games with the quadratic hedge as well as more general weaker hedges. The latter corresponds to existence of an absolute moment of order smaller than two in the measure-theoretic framework. We prove some precise relations between the convergence of centered random series and the convergence of the series of prices of the hedges. When interpreted in measure-theoretic framework, these results characterize convergence of a martingale in terms of convergence of the series of conditional absolute moments. In order to prove these results we derive some fundamental results on deterministic strategies of Reality, who is a player in a protocol of game-theoretic probability. It is of particular interest, since Reality's strategies do not have any counterparts in measure-theoretic framework, ant yet they can be used to prove results which can be interpreted in measure-theoretic framework.

We prove both the validity and the sharpness of the law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges.O( A n / ln ln A n ) and which does not obey the LIL. A number of other sufficient conditions for the LIL (1) were given in the literature such as [2,8]. For instance, Egorov [1] showed that the following is a sufficient condition:for any ǫ > 0. Our result gives a new sufficient condition (2) for the LIL (1).In the case of independent, identically distributed (i.i.d.) random variables, Hartman and Wintner [3] proved that existence of a second moment suffices for the LIL and Strassen [10] proved conversely that existence of a second moment is necessary. The topic of this paper is the LIL in game-theoretic probability, which was studied in Shafer and Vovk [9] under two protocols. The first protocol "unbounded forecasting" only contains a quadratic hedge.