Convergence of random walks on the circle generated by an irrational rotation

Abstract

Abstract. Fix α ∈ [0, 1). Consider the random walk on the circle S 1 which proceeds by repeatedly rotating points forward or backward, with probability , by an angle 2πα. This paper analyzes the rate of convergence of this walk to the uniform distribution under "discrepancy" distance. The rate depends on the continued fraction properties of the number ξ = 2α. We obtain bounds for rates when ξ is any irrational, and a sharp rate when ξ is a quadratic irrational. In that case the discrepancy falls as k − 1 2 (up to constant factors), where k is the number of steps in the walk. This is the first example of a sharp rate for a discrete walk on a continuous state space. It is obtained by establishing an interesting recurrence relation for the distribution of multiples of ξ which allows for tighter bounds on terms which appear in the Erdős-Turán inequality.

On the discrepancy of random subsequences of $\{nα\}$ II

“…N (S k α) is the empirical analogue of ψ α (k). The order of magnitude of ψ α (k) was investigated by Schatte [20], Diaconis [12], Su [24], Hensley and Su [15]; improving their results, in [3] we showed that…”

“…Further, M ℓ = 2 n + m ℓ 2 ℓ for some integer 0 ≤ m ℓ < 2 n−ℓ . Applying (24) on the interval [1, 2 n ] with H = n/2 and on the intervals [M ℓ + 1, M ℓ + 2 ℓ−1 ] for all n/2 ≤ ℓ ≤ n such that β ℓ−1 = 1 with the choice H = 2 ℓ/2 , we get that…”

Diophantine Gaussian excursions and random walks

“…It is clear that if ω's coordinates are well approximable by rationals, the same goes for the increments of the random walk, hence it is likely to come back closer to 0 faster. The study of random walks on a group started on finite arithmetic groups with the works of Diaconis, Saloff-Coste, Rosenthal, Porod, (see references in [24]) and results for such irrational random walks in the continuous settings were then achieved by Diaconis [8], and finally Su [24], who gave the optimal speed of convergence of the law of U n in an appropriate distance. Then Prescott and Su [22] extended the study in higher dimensional tori.…”

“…According to the Central Limit Theorem, the law of the renormalised sum n −1/2 U n weakly converges to a Gaussian measure (see also Lemma 19 for precise estimates), and the law µ n of U n is known to converge to Lebesgue measure on [0, 1[ d [24]. But it seems that if we zoom in further on this convergence, it becomes very irregular.…”

A LeVeque-type lower bound for discrepancy

“…We give an example at the conclusion of this paper. See [9] and [10] for further work in this direction.…”