Invariance Properties of Random Sequences

We present invariance characterizations of di erent types of random sequences. We correct Schnorr's original, incorrect characterization of Martin-L of random sequences, compare it with Schnorr's corresponding characterization of his own randomness concept, and give a similar, new chararacterization of Kurtz random sequences. That is, we show that an in nite sequence is Kurtz random if and only if for every partial, computable, measure-invariant function : ! ! ! the sequence ( ) is not recursive.