The asymptotic distribution of the remainder in a certain base-$β$ expansion

Ira W. Herbst, Jesper Møller, Anne Marie Svane

Published 2023-12-15Version 1

Let $X=\sum_{k=1}^\infty X_k \beta^{-k}$ be the base-$\beta$ expansion of a continuous random variable $X$ on the unit interval where $\beta$ is the golden ratio. We study the asymptotic distribution and convergence rate of the scaled remainder $\sum_{k=n+1}^\infty X_k \beta^{n-k}$ when $n$ tends to infinity.