The Smallest Probability Interval a Sequence Is Random for: A Study for Six Types of Randomness

There are many randomness notions. On the classical account, many of them are about whether a given infinite binary sequence is random for some given probability. If so, this probability turns out to be the same for all these notions, so comparing them amounts to finding out for which of them a given sequence is random. This changes completely when we consider randomness with respect to probability intervals, because here, a sequence is always random for at least one interval, so the question is not if, but rather for which intervals, a sequence is random. We show that for many randomness notions, every sequence has a smallest interval it is (almost) random for. We study such smallest intervals and use them to compare a number of randomness notions. We establish conditions under which such smallest intervals coincide, and provide examples where they do not.