A Central-Limit-Theorem for Isotropic Random-Walks on n-Spheres for n → ∞

Let (Yl)l≥0 be an isotropic random walk on the n-sphere Sn ⊂ Rn+1 starting at x0 ∈ Sn. Then the random variables Xl ≔ cos ∠(Yl, x0) form a Markov chain on [−1, 1] whose transition probabilities are closely related to ultraspherical convolutions on [−1, 1]. We prove that √nXl is normally distributed for n, l → ∞ provided that the spherical distances of the jumps of Yl tend to 0. The expectation of this distribution depends on relations between n, l and the spherical distances of the random jumps.