  Why should this be a problem for random sequences? The property of large numbers shows that almost all infinite binary sequences have limit digit frequency \(\frac{1}{2}\), but says nothing about how quickly this convergence happens or about the statistical properties of the initial segments. There certainly exist sequences that converge to \(\frac{1}{2}\) but where \(S_n (x)/n \gt \frac{1}{2}\) for all \(n\) (the sequence has the right mean but converges ‘from above’). In the ‘random walk’ model of our random sequences, where each element in the sequence is interpreted as a step left (if 0) or right (if 1) along the integer line, this sequence would consist of a walk that (in the limit) ends up back at the origin but always (or even eventually) stays to the right. Intuitively, such a sequence is not random.