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A Stein characterisation of the distribution of the product of correlated normal random variables

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  • Gaunt, Robert E.
  • Li, Siqi
  • Sutcliffe, Heather L.

Abstract

We obtain a Stein characterisation of the distribution of the product of two correlated normal random variables with non-zero means, and more generally the distribution of the sum of independent copies of such random variables. Our Stein characterisation is shown to naturally generalise a number of other Stein characterisations in the literature. From our Stein characterisation we derive recursive formulas for the moments of the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, which allows for efficient computation of higher order moments.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:stapro:v:216:y:2025:i:c:s0167715224002384
    DOI: 10.1016/j.spl.2024.110269
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    References listed on IDEAS

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    1. Gaunt, Robert E., 2023. "On the moments of the variance-gamma distribution," Statistics & Probability Letters, Elsevier, vol. 201(C).
    2. 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.
    3. Antonio Seijas-Macias & Amílcar Oliveira & Teresa A. Oliveira, 2023. "The skewness and kurtosis of the product of two normally distributed random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 52(1), pages 80-93, January.
    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. Robert E. Gaunt, 2020. "Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I," Journal of Theoretical Probability, Springer, vol. 33(1), pages 465-505, March.
    6. 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.
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