  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.
