Limit theorems for counting large continued fraction digits

Abstract

We establish a central limit theorem for counting large continued fraction digits $(a_n)$, i.e. we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp which additionally assumes that $b_n$ tends to infinity. Moreover, we give a refinement of the famous Borel-Bernstein Theorem for continued fractions regarding the event that the $n$-th continued fraction digit lies infinitely often between $d_n$ and $d_n(1+1/c_n)$ for given sequences $(c_n)$ and $(d_n)$. Also for these sets we obtain a central limit theorem. As an interesting side result we determine the first $\varphi$-mixing coefficient for the Gauss system explicitly.

14 pages

References in corpus (2)

• Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of $E_2$
• Upper and lower fast Khintchine spectra in continued fractions

Cited by in corpus (1)

• On the continued fraction expansion of absolutely normal numbers