A completion of counterexamples to the classical central limit theorem for triplewise independent and identically distributed random variables

Raič, Martin (author)

By the Lindeberg–Lévy central limit theorem, standardized partial sums of a sequence of mutually independent and identically distributed random variables converge in law to the standard normal distribution. It is known that mutual independence cannot be relaxed to pairwise independence, nor even to triplewise independence. Counterexamples have been constructed for most marginal distributions: a recent construction works under a condition which excludes certain probability distributions with atomic parts, in particular almost all distributions on a fixed finite set. In the present paper, we show that this condition can be lifted: for any probability distribution ▫$F$▫ on the real line, which has finite variance and is not concentrated in a single point, there exists a sequence of triplewise independent random variables with distribution ▫$F$▫, such that its standardized partial sums converge in law to a distribution which is not normal. There is also scope for extension to ▫$k$▫-tuplewise independence.

• Language: English
• Keywords: central limit theorem, mutual independence, triplewise independence
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: 5 str.
• Numbering: Vol. 226, art. 110508
• PID: 20.500.12556/RUL-175454
• UDC: 519.2
• ISSN on article: 0167-7152
• DOI: 10.1016/j.spl.2025.110508
• COBISS.SI-ID: 245761283
• Publication date in RUL: 28.10.2025

Record is a part of a journal

• Title: Statistics & probability letters
• Shortened title: Stat. probab. lett.
• Publisher: Elsevier
• ISSN: 0167-7152
• COBISS.SI-ID: 1248533

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: centralni limitni izrek, vzajemna neodvisnost, neodvisnost po trojicah

Projects

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P1-0448
• Name: Stohastične metode in njihova uporaba