Central Limit Theorems for Random Walks on the 3-Regular Tree

Abstract

We present four related limit theorems for random walks, beginning with a one-dimensional nearest-neighbour walk and its absolute value. The absolute value has a normal limit after centring when the drift is nonzero, and a half-normal limit when the drift is zero. The main part of the note concerns simple random walk on the 3-regular tree. A coupling gives the distance central limit theorem, and conditional uniformity of reduced words leads to a central limit theorem for the number of occurrences of a fixed letter in the endpoint. The latter count, centred at n/9 and divided by √n, converges to a normal law with variance 10/81. We compute the fixed-length covariance explicitly and establish the joint convergence that separates the limiting variance into two asymptotically independent contributions: 2/81 from the letter composition and 8/81 from the word length.