Let $X=\sum_{k=1}^\infty X_k β^{-k}$ be the base-$β$ expansion of a continuous random variable $X$ on the unit interval where $β$ is the positive solution to $β^n = 1 + β+ \cdots + β^{n-1}$ for an integer $n\ge 2$ (i.e., $β$ is a generalization of the golden mean for which $n=2$). We study the asymptotic distribution and convergence rate of the scaled remainder $\sum_{k=1}^\infty X_{m+k} β^{-k}$ when $m$ tends to infinity.
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