The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable

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