Kac’s central limit theorem by Stein’s method

Suprio Bhar, Ritwik Mukherjee and Prathmesh Patil

Statistics & Probability Letters, 2025, vol. 219, issue C

Abstract: In 1946, Mark Kac proved a Central Limit type theorem for a sequence of random variables that were not independent. The random variables under consideration were obtained from the angle-doubling map. The idea behind Kac’s proof was to show that although the random variables under consideration were not independent, they were what he calls statistically independent (in modern terminology, this concept is called long range independence). Using that observation, Kac showed that the sample averages of the random variables, suitably normalized, converges to the standard normal distribution. In this paper, we give a new proof of Kac’s result by applying Stein’s method. We show that the normalized sample averages converge to the standard normal distribution in the Wasserstein metric, which in particular implies convergence in distribution.

Keywords: Central limit theorem; Wasserstein metric; Convergence in distribution; Angle-doubling map; Stein’s method (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0167715224002980
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:stapro:v:219:y:2025:i:c:s0167715224002980

Ordering information: This journal article can be ordered from
http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spl.2024.110329

Access Statistics for this article

Statistics & Probability Letters is currently edited by Somnath Datta and Hira L. Koul

More articles in Statistics & Probability Letters from Elsevier
Bibliographic data for series maintained by Catherine Liu ().