### abstract ###
Solomonoff's central result on induction is that the prediction of a universal semimeasure  SYMBOL  converges rapidly and with probability 1 to the true sequence generating predictor  SYMBOL , if the latter is computable
Hence,  SYMBOL  is eligible as a universal sequence predictor in case of unknown  SYMBOL
Despite some nearby results and proofs in the literature, the stronger result of convergence for all (Martin-L{\"o}f) random sequences remained open
Such a convergence result would be particularly interesting and natural, since randomness can be defined in terms of  SYMBOL  itself
We show that there are universal semimeasures  SYMBOL  which do not converge to  SYMBOL  on all  SYMBOL -random sequences, ie \ we give a partial negative answer to the open problem
We also provide a positive answer for some non-universal semimeasures
We define the incomputable measure  SYMBOL  as a mixture over all computable measures and the enumerable semimeasure  SYMBOL  as a mixture over all enumerable nearly-measures
We show that  SYMBOL  converges to  SYMBOL  and  SYMBOL  to  SYMBOL  on all random sequences
The Hellinger distance measuring closeness of two distributions plays a central role
### introduction ###
A sequence prediction task is defined as to predict the next symbol  SYMBOL  from an observed sequence  SYMBOL
The key concept to attack general prediction problems is Occam's razor, and to a less extent Epicurus' principle of multiple explanations
The former/latter may be interpreted as to keep the simplest/all theories consistent with the observations  SYMBOL  and to use these theories to predict  SYMBOL
Solomonoff  CITATION  formalized and combined both principles in his universal a priori semimeasure  SYMBOL  which assigns high/low probability to simple/complex environments  SYMBOL , hence implementing Occam and Epicurus
Formally it can be represented as a mixture of all enumerable semimeasures
An abstract characterization of  SYMBOL  by Levin  CITATION  is that  SYMBOL  is a universal enumerable semimeasure in the sense that it multiplicatively dominates all enumerable semimeasures
Solomonoff's  CITATION  central result is that if the probability  SYMBOL  of observing  SYMBOL  at time  SYMBOL , given past observations  SYMBOL  is a computable function, then the universal predictor  SYMBOL  converges (rapidly )  with  SYMBOL -probability 1  (w p 1) for  SYMBOL  to the optimal/true/informed predictor  SYMBOL , hence  SYMBOL  represents a universal predictor in case of unknown ``true'' distribution  SYMBOL
Convergence of  SYMBOL  to  SYMBOL  w p 1 tells us that  SYMBOL  is close to  SYMBOL  for sufficiently large  SYMBOL  for almost all sequences  SYMBOL
It says nothing about whether convergence is true for any  particular  sequence (of measure 0)
Martin-L{\"o}f (M L ) randomness is the standard notion for randomness of individual sequences  CITATION
A M L -random sequence passes  all  thinkable effective randomness tests, eg \ the law of large numbers, the law of the iterated logarithm, etc
In particular, the set of all  SYMBOL -random sequences has  SYMBOL -measure 1
It is natural to ask whether  SYMBOL  converges to  SYMBOL  (in difference or ratio) individually for all M L -random sequences
Clearly, Solomonoff's result shows that convergence may at most fail for a set of sequences with  SYMBOL -measure zero
A convergence result for M L -random sequences would be particularly interesting and natural in this context, since M L -randomness can be defined in terms of  SYMBOL  itself  CITATION
Despite several attempts to solve this problem  CITATION , it remained open  CITATION
In this paper we construct an M L -random sequence and show the existence of a universal semimeasure which does not converge on this sequence, hence answering the open question negatively for some  SYMBOL
It remains open whether there exist (other) universal semimeasures, probably with particularly interesting additional structure and properties, for which M L -convergence holds
The main positive contribution of this work is the construction of a non-universal enumerable semimeasure  SYMBOL  which M L -converges to  SYMBOL  as desired
As an intermediate step we consider the incomputable measure  SYMBOL , defined as a mixture over all computable measures
We show M L -convergence of predictor  SYMBOL  to  SYMBOL  and of  SYMBOL  to  SYMBOL
The Hellinger distance measuring closeness of two predictive distributions plays a central role in this work
The paper is organized as follows: In Section~ we give basic notation and results (for strings, numbers, sets, functions, asymptotics, computability concepts, prefix Kolmogorov complexity), and define and discuss the concepts of (universal) (enumerable) (semi)measures
Section~ summarizes Solomonoff's and G\'acs' results on predictive convergence of  SYMBOL  to  SYMBOL  with probability 1
Both results can be derived from a bound on the expected Hellinger sum
We present an improved bound on the expected exponentiated Hellinger sum, which implies very strong assertions on the convergence rate
In Section~ we investigate whether convergence for all Martin-L{\"o}f random sequences hold
We construct a  SYMBOL -M L -random sequence on which some universal semimeasures  SYMBOL  do not converge to  SYMBOL
We give a non-constructive and a constructive proof of different virtue
In Section~ we present our main positive result
We derive a finite bound on the Hellinger sum between  SYMBOL  and  SYMBOL , which is exponential in the randomness deficiency of the sequence and double exponential in the complexity of  SYMBOL
This implies that the predictor  SYMBOL  M L -converges to  SYMBOL
Finally, in Section~ we show that  SYMBOL  is non-universal and asymptotically M L -converges to  SYMBOL , and summarize the computability, measure, and dominance properties of  SYMBOL ,  SYMBOL ,  SYMBOL , and  SYMBOL
Section~ contains discussion and outlook
