Convergence of Random Walks on the Circle Generated by an Irrational Rotation
نویسنده
چکیده
Fix α ∈ [0, 1). Consider the random walk on the circle S1 which proceeds by repeatedly rotating points forward or backward, with probability 1 2 , 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.
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