For i.i.d. uniform random directions on the sphere, the greedy sign choice has a unique invariant radial measure with mean sqrt(pi)/2 * Gamma((d+1)/2)/Gamma(d/2) ~ sqrt(pi d/8).
Harris (1956) The existence of stationary measures for certain Markov processes
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Approximately Jumping Towards the Origin
For i.i.d. uniform random directions on the sphere, the greedy sign choice has a unique invariant radial measure with mean sqrt(pi)/2 * Gamma((d+1)/2)/Gamma(d/2) ~ sqrt(pi d/8).