Large deviation principle for arithmetic functions in continued fraction expansion
Large deviation principle for arithmetic functions in continued fraction expansion
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DOI：
10.1007/s00605-019-01322-5
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发表时间：
2019-07
期刊：
影响因子：
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通讯作者：
Hiroki Takahasi
中科院分区：
文献类型：
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作者：
Hiroki Takahasi
Khinchin证明，算术平均经常连续分数数字勒贝格几乎每一个无理数在（0，1）发散到无穷大。因此，没有一个经典的极限定理，如弱和强大数定律或中心极限定理成立。然而，我们证明了存在一个大的偏差率函数，估计指数概率的算术平均数远离无穷大。这导致我们与广泛认同的观点相矛盾，即大偏差原理是大数定律的细化：前者可能比后者更普遍。
Khinchin proved that the arithmetic mean of the regular continued fraction digits of Lebesgue almost every irrational number in (0, 1) diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers or central limit theorems hold. Nevertheless, we prove the existence of a large deviations rate function which estimates exponential probabilities with which the arithmetic mean of digits stays away from infinity. This leads us to a contradiction to the widely-shared view that the large deviation principle is a refinement of laws of large numbers: the former can be more universal than the latter.