On the product of correlated normal random variables and the noncentral chi-square difference distribution
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914020907286528 |
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| author | Gaunt, Robert E. |
| author_facet | Gaunt, Robert E. |
| contents | We represent the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, as a difference of two independent noncentral chi-square random variables (which we refer to as the noncentral chi-square difference distribution). As a consequence, we obtain, amongst other results, an exact formula for the probability density function of the noncentral chi-square difference distribution, a Stein characterisation of the noncentral chi-square difference distribution, a simple formula for the moments of the sum of independent copies of the product of correlated normal random variables, an exact formula for the probability that such a random variable is negative, and also show that such random variables are self-decomposable and provide a Lévy-Khintchine representation of the characteristic function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_04101 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the product of correlated normal random variables and the noncentral chi-square difference distribution Gaunt, Robert E. Probability Primary 60E05, 62E15 We represent the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, as a difference of two independent noncentral chi-square random variables (which we refer to as the noncentral chi-square difference distribution). As a consequence, we obtain, amongst other results, an exact formula for the probability density function of the noncentral chi-square difference distribution, a Stein characterisation of the noncentral chi-square difference distribution, a simple formula for the moments of the sum of independent copies of the product of correlated normal random variables, an exact formula for the probability that such a random variable is negative, and also show that such random variables are self-decomposable and provide a Lévy-Khintchine representation of the characteristic function. |
| title | On the product of correlated normal random variables and the noncentral chi-square difference distribution |
| topic | Probability Primary 60E05, 62E15 |
| url | https://arxiv.org/abs/2408.04101 |