We establish a central limit theorem for counting large continued fraction digits ( a n ), that is, 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 explicitly determine the first φ-mixing coefficient for the Gauss system.

中文翻译：

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计算大连分数的极限定理

我们建立了一个计算大连分数数字 (an) 的中心极限定理，即我们计算出现次数 { an >bn }，其中 ( bn ) 是一个正整数序列。我们的结果改进了 Philipp 的类似结果，该结果还假设 bn 趋于无穷大。此外，对于给定的序列 (cn) 和 (dn)，关于第 n 个连分数位无限常位于 dn 和 dn (1 + 1/ cn ) 之间的事件，我们对著名的连分数 Borel-Bernstein 定理进行了改进. 此外，对于这些集合，我们获得了中心极限定理。作为一个有趣的附带结果，我们明确地确定了高斯系统的第一个 φ 混合系数。