  Solomonoff's central result on induction is that the posterior of a universal
semimeasure M converges rapidly and with probability 1 to the true sequence
generating posterior mu, if the latter is computable. Hence, M is eligible as a
universal sequence predictor in case of unknown mu. Despite some nearby results
and proofs in the literature, the stronger result of convergence for all
(Martin-Loef) random sequences remained open. Such a convergence result would
be particularly interesting and natural, since randomness can be defined in
terms of M itself. We show that there are universal semimeasures M which do not
converge for all random sequences, i.e. 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 D as a mixture over all
computable measures and the enumerable semimeasure W as a mixture over all
enumerable nearly-measures. We show that W converges to D and D to mu on all
random sequences. The Hellinger distance measuring closeness of two
distributions plays a central role.
