| Title: | A completion of counterexamples to the classical central limit theorem for triplewise independent and identically distributed random variables |
|---|
| Authors: | ID Raič, Martin (Author) |
|---|
| Files: |  PDF - Presentation file, download (641,42 KB)
MD5: 4067F46B6369DE8B1C15FA2DC04FA44F
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0167715225001531
|
|---|
| Language: | English |
|---|
| Typology: | 1.01 - Original Scientific Article |
|---|
| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
|
|---|
| Abstract: | 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. |
|---|
| Keywords: | central limit theorem, mutual independence, triplewise independence |
|---|
| Publication status: | Published |
|---|
| Publication version: | Version of Record |
|---|
| Publication date: | 01.11.2025 |
|---|
| Year of publishing: | 2025 |
|---|
| Number of pages: | 5 str. |
|---|
| Numbering: | Vol. 226, article no. 110508 |
|---|
| PID: | 20.500.12556/DiRROS-23328  |
|---|
| UDC: | 519.2 |
|---|
| ISSN on article: | 0167-7152 |
|---|
| DOI: | 10.1016/j.spl.2025.110508 |
|---|
| COBISS.SI-ID: | 245761283 |
|---|
| Publication date in DiRROS: | 18.08.2025 |
|---|
| Views: | 296 |
|---|
| Downloads: | 146 |
|---|
| Metadata: |    |
|---|
|
:
|
Copy citation |
|---|
| | | | Share: |  |
|---|
Hover the mouse pointer over a document title to show the abstract or click
on the title to get all document metadata. |
