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The basic distributional theory for the product of zero mean correlated normal random variables

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  • Robert E. Gaunt

Abstract

The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener‐Itô integrals.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:stanee:v:76:y:2022:i:4:p:450-470
    DOI: 10.1111/stan.12267
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    References listed on IDEAS

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    1. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
    2. Robert E. Gaunt, 2021. "Stein’s method and the distribution of the product of zero mean correlated normal random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(2), pages 280-285, January.
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    Cited by:

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