A note on the equivalence between the conditional uncorrelation and the independence of random variables
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929319653146624 |
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| author | Jaworski, Piotr Jelito, Damian Pitera, Marcin |
| author_facet | Jaworski, Piotr Jelito, Damian Pitera, Marcin |
| contents | It is well known that while the independence of random variables implies zero correlation, the opposite is not true. Namely, uncorrelated random variables are not necessarily independent. In this note we show that the implication could be reversed if we consider the localised version of the correlation coefficient. More specifically, we show that if random variables are conditionally (locally) uncorrelated for any quantile conditioning sets, then they are independent. For simplicity, we focus on the absolutely continuous case. Also, we illustrate potential usefulness of the stated result using two simple examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_16655 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A note on the equivalence between the conditional uncorrelation and the independence of random variables Jaworski, Piotr Jelito, Damian Pitera, Marcin Statistics Theory Probability 60E05, 62E10, 62H20 It is well known that while the independence of random variables implies zero correlation, the opposite is not true. Namely, uncorrelated random variables are not necessarily independent. In this note we show that the implication could be reversed if we consider the localised version of the correlation coefficient. More specifically, we show that if random variables are conditionally (locally) uncorrelated for any quantile conditioning sets, then they are independent. For simplicity, we focus on the absolutely continuous case. Also, we illustrate potential usefulness of the stated result using two simple examples. |
| title | A note on the equivalence between the conditional uncorrelation and the independence of random variables |
| topic | Statistics Theory Probability 60E05, 62E10, 62H20 |
| url | https://arxiv.org/abs/2210.16655 |