The Asymptotic Distribution of the Scaled Remainder for Pseudo Golden Ratio Expansions of a Continuous Random Variable - Herbst, Ira W.; Møller, Jesper; Svane, Anne Marie (Methodology and computing in applied probability, 2025)
Let X = ∑ k = 1 ∞ X k β - k be the (greedy) base- β expansion of a continuous random variable X on the unit interval where β is the positive solution to β n = 1 + β + ⋯ + β n - 1 for an integer n ⩾ 2 (i.e., β is a generalization of the golden mean corresponding to n = 2 ). We study the asymptotic distribution and convergence rate of the scaled remainder ∑ k = 1 ∞ X m + k β - k when m tends to infinity.