Limit theorems related to beta-expansion and continued fraction expansion

Let $\beta >1$ be a real number and $x\in [0,1)$ be an irrational number. Denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$ ($n\in \mathbb{N}$). In this paper, we show a central limit theorem and a law of the iterated logarithm for the random variables sequence $\{k_n,n\ge 1\}$, which generalize the results of Faivre and Wu respectively from $\beta =10$ to any $\beta >1$.