The complexity of stochastic sequences

Abstract

We observe that known results on the Kolmogorov complexity of prefixes of effectively stochastic sequences extend to corresponding random sequences. First, there are recursively random sequences such that for any nondecreasing and unbounded computable function f and for almost all n, the uniform complexity of the length n prefix of the sequence is bounded by f(n). Second, a similar result with bounds of the form f(n) log n holds for partially-recursive random sequences. Furthermore, we show that there is no Mises-Wald-Church stochastic sequence such that the prefixes of the sequence have Kolmogorov complexity O(log n). This result implies a sharp bound for the complexity of the prefixes of Mises-Wald-Church stochastic and of partially-recursive random sequences. As an immediate corollary to our results, we obtain the known separation of the classes of recursively random and of Mises-Wald-Church stochastic sequences.