### Abstract

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.

Original language | English |
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Journal | Monatshefte fur Mathematik |

DOIs | |

Publication status | Published - 2019 Jan 1 |

### Fingerprint

### Keywords

- Arithmetic mean of digits
- Continued fraction
- Large deviation principle

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Large deviation principle for arithmetic functions in continued fraction expansion.** / Takahasi, Hiroki.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Large deviation principle for arithmetic functions in continued fraction expansion

AU - Takahasi, Hiroki

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Arithmetic mean of digits

KW - Continued fraction

KW - Large deviation principle

UR - http://www.scopus.com/inward/record.url?scp=85069475226&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85069475226&partnerID=8YFLogxK

U2 - 10.1007/s00605-019-01322-5

DO - 10.1007/s00605-019-01322-5

M3 - Article

AN - SCOPUS:85069475226

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

ER -