Another note on ???Euclidean algorithms are Gaussian??? by V. Baladi and B. Vall??e

We record two remarks on the work of Baladi???Vall??e [J. Number Theory 110 (2005), 331???386]. They proved the asymptotic Gaussian distribution of the length of continued fractions as a random variable on the set of rational numbers whose denominators are less than or equal to a fixed positive integer with uniform probability. We give a direct proof of that result without the smoothing process by applying Perron???s formula with error terms, and further derive an equivalent result on a thinner probability space