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
Pages (from-to)137-152
Number of pages16
JournalMonatshefte fur Mathematik
Volume190
Issue number1
DOIs
Publication statusPublished - 2019 Sep 1

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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