The Asymptotic Distribution of the Scaled Remainder for Pseudo Golden Ratio Expansions of a Continuous Random Variable

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