Large deviation principle for arithmetic functions in continued fraction expansion

Research output: Contribution to journalArticle

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 languageEnglish
JournalMonatshefte fur Mathematik
DOIs
Publication statusPublished - 2019 Jan 1

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Arithmetic Functions
Continued Fraction Expansion
Large Deviation Principle
Digit
Infinity
Weak law of large numbers
Irrational number
Strong law of large numbers
Classical Limit
Rate Function
Law of large numbers
Henri Léon Lebésgue
Continued fraction
Diverge
Limit Theorems
Large Deviations
Central limit theorem
Refinement
Estimate

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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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.

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