A random walker can choose the distance of each step but not the direction, which is continuous in q dimensions. The walker, who can initially be anywhere in q-space, wants to get ever-closer to the origin, in the sense that by taking enough steps he can be arbitrarily certain of being arbitrarily close. The walker travels, at each step t, a distance dt = f(Ot-1,q) where Ot-1 is the distance from the origin at the end of step t-1.
I postulate that choosing dt = Ot-1 will result in a path which converges on the origin for all q, and that no other choice will converge faster. Either prove this, or find the minimum value of q for which the path will not converge.
Is there a value of q for which (dt = Ot-1) converges, but no other choice of dt = f(Ot-1,q) converges?