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

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 golden ratio. We study the asymptotic distribution and convergence rate of the scaled remainder ${\sum }_{k=n+1}^{\mathrm{\infty }}{X}_k{\beta }^{n-k}$ when $n$ tends to infinity.