[vladimirralev]

They say "these results are completely general for any probability distribution with zero mean and a finite covariance matrix with rank much larger than the number of steps". It's not clear to me if that condition implies the number of steps is much lower than the dimensions of the random walk space or perhaps the probability distribution needs to be concentrated into a smaller number of dimensions to begin with? In which case the results is much less shocking.

[antognini]

The condition is the former. The probability distribution spans the full dimensionality of the space. Basically, the result will hold for an infinite number of dimensions and a finite number of steps. But it will also hold if you take both the number of steps and the dimensionality to infinity while holding the ratio N_steps / D constant with N_steps / D << 1.