  Such sequences do indeed violate at least one property of stochasticity, as it turns out that in a measure one set of sequences, \(S_n (x)/n\) will be above the mean infinitely many times, and below the mean infinitely many times. So stated, this is the law of symmetric oscillation (Dasgupta, 2011:  13).[10]  Since the law of symmetric oscillations holds for a measure one set of sequences, it is a plausible property of randomness (it is naturally of a family with other properties of stochasticity). Ville’s result shows that von Mises’ definition in terms of place selections cannot characterise random sequences exactly because it includes sequences that violate this law (so don’t correspond to a truly random random walk). Indeed, such sequences don’t even correspond to von Mises’ avowed aims. As Li and Vitányi say (2008: 54), ‘if you bet 1 all the time against such a sequence of outcomes, your accumulated gain is always positive’. As such, Ville-style sequences seem to permit successful gambling, despite the fact that they do not permit a system to be formulated in terms of place selections.