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On the product of correlated normal random variables and the noncentral chi-square difference distribution

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  • Gaunt, Robert E.

Abstract

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.

Suggested Citation

  • Gaunt, Robert E., 2026. "On the product of correlated normal random variables and the noncentral chi-square difference distribution," Statistics & Probability Letters, Elsevier, vol. 227(C).
    • Handle: RePEc:eee:stapro:v:227:y:2026:i:c:s0167715225001993
    DOI: 10.1016/j.spl.2025.110554
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    References listed on IDEAS

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    1. Robert E. Gaunt, 2019. "A note on the distribution of the product of zero‐mean correlated normal random variables," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 73(2), pages 176-179, May.
    2. Gaunt, Robert E., 2023. "On the moments of the variance-gamma distribution," Statistics & Probability Letters, Elsevier, vol. 201(C).
    3. Gaunt, Robert E. & Li, Siqi & Sutcliffe, Heather L., 2025. "A Stein characterisation of the distribution of the product of correlated normal random variables," Statistics & Probability Letters, Elsevier, vol. 216(C).
    4. Robert E. Gaunt, 2022. "The basic distributional theory for the product of zero mean correlated normal random variables," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 76(4), pages 450-470, November.
    5. Forrester, Peter J., 2024. "On the gamma difference distribution," Statistics & Probability Letters, Elsevier, vol. 211(C).
    6. Gaunt, Robert E., 2019. "Stein operators for variables form the third and fourth Wiener chaoses," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 118-126.
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    Cited by:

    1. Jones, M.C., 2026. "On mixture relationships between central and non-central chi-squared difference distributions," Statistics & Probability Letters, Elsevier, vol. 229(C).

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