Probability theory/Statistics
On the distribution of the product of correlated normal random variables
[Sur la distribution exacte du produit de variables aléatoires normales corrélées]
Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 201-204.

Dans cette Note, on résout un problème, posé depuis 1936, sur la distribution exacte du produit de variables aléatoires normales corrélées. Comme résultat supplémentaire, on déduit la distribution exacte de la moyenne du produit de variables aléatoires normales corrélées.

We solve a problem that has remained unsolved since 1936 – the exact distribution of the product of two correlated normal random variables. As a by-product, we derive the exact distribution of the mean of the product of correlated normal random variables.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.10.019
Nadarajah, Saralees 1 ; Pogány, Tibor K. 2

1 School of Mathematics, University of Manchester, Manchester M13 9PL, UK
2 Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Croatia
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Nadarajah, Saralees; Pogány, Tibor K. On the distribution of the product of correlated normal random variables. Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 201-204. doi : 10.1016/j.crma.2015.10.019. http://www.numdam.org/articles/10.1016/j.crma.2015.10.019/

[1] Aroian, L.A. The probability function of the product of two normally distributed variables, Ann. Math. Stat., Volume 18 (1944), pp. 265-271

[2] Aroian, L.A.; Taneja, V.S.; Cornwell, L.W. Mathematical forms of the distribution of the product of two normal variables, Commun. Stat., Theory Methods, Volume 7 (1978), pp. 165-172

[3] Bandi, M.M.; Connaughton, C. Craig's XY distribution and the statistics of Lagrangian power in two-dimensional turbulence, Phys. Rev. E, Volume 77 (2008)

[4] Craig, C.C. On the frequency function of xy, Ann. Math. Stat., Volume 7 (1936), pp. 1-15

[5] Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products, Academic Press, San Diego, CA, USA, 2000

[6] Haldane, J.B.S. Moments of the distributions of powers and products of normal variates, Biometrika, Volume 32 (1942), pp. 226-242

[7] Hayya, J.C.; Ferrara, W.L. On normal approximations of the frequency functions of standard forms where the main variables are normally distributed, Manag. Sci., Volume 19 (1972), pp. 173-186

[8] MacKinnon, D. Introduction to Statistical Mediation Analysis, Routledge, New York, 2012

[9] MacKinnon, D.P.; Fritz, M.S.; Williams, J.; Lockwood, C.M. Distribution of the product confidence limits for the indirect effect: program PRODCLIN, Behav. Res. Methods, Volume 39 (2007), pp. 384-389

[10] Meeker, W.Q.; Odeh, R.D.; Cornwell, L.W.; Aroian, L.A.; Kennedy, W.J. Selected Tables in Mathematical Statistics: The Product of Two Normally Distributed Random Variables, American Mathematical Society, Providence, RI, USA, 1981

[11] Ware, R.; Lad, F. Approximating the distribution for sums of products of normal variables, Department of Mathematics and Statistics, University of Canterbury, New Zealand, 2013 (Working paper)

  • Gaunt, Robert E. Absolute moments of the variance-gamma distribution, Journal of Mathematical Analysis and Applications, Volume 543 (2025) no. 1, p. 11 (Id/No 128861) | DOI:10.1016/j.jmaa.2024.128861 | Zbl:7946870
  • Gaunt, Robert E.; Ye, Zixin Asymptotic approximations for the distribution of the product of correlated normal random variables, Journal of Mathematical Analysis and Applications, Volume 543 (2025) no. 2, p. 128987 | DOI:10.1016/j.jmaa.2024.128987
  • Gaunt, Robert E.; Li, Siqi; Sutcliffe, Heather L. A Stein characterisation of the distribution of the product of correlated normal random variables, Statistics Probability Letters, Volume 216 (2025), p. 10 (Id/No 110269) | DOI:10.1016/j.spl.2024.110269 | Zbl:7955926
  • Stutzer, N.-O.; Lunde, J. G. S.; Breysse, P. C.; Chung, D. T.; Cleary, K. A.; Dunne, D. A.; Eriksen, H. K.; Ihle, H. T.; Padmanabhan, H.; Tolgay, D.; Wehus, I. K.; Bond, J. R.; Church, S. E.; Gaier, T.; Gundersen, J. O.; Harris, A. I.; Harper, S. E.; Hobbs, R.; Kim, J.; Lamb, J. W.; Lawrence, C. R.; Murray, N.; Pearson, T. J.; Philip, L.; Readhead, A. C. S.; Rennie, T. J.; Woody, D. P. COMAP Pathfinder – Season 2 results, Astronomy Astrophysics, Volume 691 (2024), p. A336 | DOI:10.1051/0004-6361/202451123
  • GAUNT, ROBERT E. ON THE CUMULATIVE DISTRIBUTION FUNCTION OF THE VARIANCE-GAMMA DISTRIBUTION, Bulletin of the Australian Mathematical Society, Volume 110 (2024) no. 2, p. 389 | DOI:10.1017/s0004972723001387
  • Ogasawara, Haruhiko The product distribution of more-than-two correlated normal variables: A didactic review with some new findings, Communications in Statistics - Theory and Methods (2024), p. 1 | DOI:10.1080/03610926.2024.2437487
  • Pang, Jie; Gao, Bo; Wang, Ning Asymptotic Spectral Distribution of a Second-Order Progressive Scattering Channel, IEEE Signal Processing Letters, Volume 31 (2024), p. 1404 | DOI:10.1109/lsp.2024.3399666
  • Rostami Ghadi, Farshad; López-Martínez, F. Javier Performance Analysis of SWIPT Relay Networks Over Arbitrary Dependent Fading Channels, IEEE Transactions on Communications, Volume 72 (2024) no. 6, p. 3651 | DOI:10.1109/tcomm.2024.3359034
  • Laumann, Timothy O.; Snyder, Abraham Z.; Gratton, Caterina Challenges in the measurement and interpretation of dynamic functional connectivity, Imaging Neuroscience, Volume 2 (2024), p. 1 | DOI:10.1162/imag_a_00366
  • Wehbeh, Jad; Sharf, Inna Nonlinear scenario-based model predictive control for quadrotors with bidirectional thrust, International Journal of Robust and Nonlinear Control, Volume 34 (2024) no. 18, pp. 12450-12475 | DOI:10.1002/rnc.7627 | Zbl:7952647
  • Gaunt, Robert E.; Li, Siqi The distribution of the product of independent variance-gamma random variables, Journal of Mathematical Analysis and Applications, Volume 539 (2024) no. 1, p. 24 (Id/No 128530) | DOI:10.1016/j.jmaa.2024.128530 | Zbl:7922124
  • Pelagatti, Matteo; Sbrana, Giacomo Estimating correlations among elliptically distributed random variables under any form of heteroskedasticity, Quantitative Finance, Volume 24 (2024) no. 3-4, pp. 451-464 | DOI:10.1080/14697688.2023.2278502 | Zbl:7885178
  • Hristov, P. O.; DiazDelaO, F. A. Subset simulation for probabilistic computer models, Applied Mathematical Modelling, Volume 120 (2023), pp. 769-785 | DOI:10.1016/j.apm.2023.03.041 | Zbl:1525.62045
  • Seijas-Macias, Antonio; Oliveira, Amílcar; Oliveira, Teresa A. The skewness and kurtosis of the product of two normally distributed random variables, Communications in Statistics. Theory and Methods, Volume 52 (2023) no. 1, pp. 80-93 | DOI:10.1080/03610926.2021.1909734 | Zbl:7649550
  • Lu, Dawei; Yang, Jingcai Identifying the distribution of linear combinations of gamma random variables via Stein's method, Communications in Statistics. Theory and Methods, Volume 52 (2023) no. 7, pp. 2116-2123 | DOI:10.1080/03610926.2021.1945104 | Zbl:7702497
  • Gaunt, Robert E.; Li, Siqi The variance-gamma ratio distribution, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 361 (2023), pp. 1151-1161 | DOI:10.5802/crmath.495 | Zbl:1525.60023
  • Wang, WenWu; Yu, Ping; Zhou, Yuejin; Tong, Tiejun; Liu, Zhonghua Equivalence of two least-squares estimators for indirect effects, Current Psychology, Volume 42 (2023) no. 9, p. 7364 | DOI:10.1007/s12144-021-02034-6
  • Leipus, Remigijus; Šiaulys, Jonas; Dirma, Mantas; Zovė, Romualdas On the distribution-tail behaviour of the product of normal random variables, Journal of Inequalities and Applications, Volume 2023 (2023), p. 13 (Id/No 32) | DOI:10.1186/s13660-023-02941-1 | Zbl:1532.60021
  • Seijas-Macias, Antonio; Oliveira, Amílcar; Oliveira, Teresa A. A New R-Function to Estimate the PDF of the Product of Two Uncorrelated Normal Variables, Mathematics, Volume 11 (2023) no. 16, p. 3515 | DOI:10.3390/math11163515
  • Hui, Jiang; Yajuan, Pan; Xiao, Wei Deviation inequalities and Cramér-type moderate deviations for non-ergodic bmα-Brownian bridge process, SCIENTIA SINICA Mathematica, Volume 53 (2023) no. 8, p. 1105 | DOI:10.1360/ssm-2021-0223
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  • Ronan, James G.; Wilson, D. Keith; Ostashev, Vladimir E. Distributions for two-point statistics of scattered signals with variable scattering strength, Waves in Random and Complex Media (2023), p. 1 | DOI:10.1080/17455030.2023.2224895
  • Gharamohammadi, Ali; Shaker, George, 2022 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (AP-S/URSI) (2022), p. 1158 | DOI:10.1109/ap-s/usnc-ursi47032.2022.9886587
  • Taylor, Christopher; Pretzner, Barbara; Zahel, Thomas; Herwig, Christoph Architectural and Technological Improvements to Integrated Bioprocess Models towards Real-Time Applications, Bioengineering, Volume 9 (2022) no. 10, p. 534 | DOI:10.3390/bioengineering9100534
  • De Oliveira, Victor; Ecker, Mark D. A non-stationary non-Gaussian hedonic spatial model for house selling prices, Communications in Statistics. Simulation and Computation, Volume 51 (2022) no. 6, pp. 2888-2905 | DOI:10.1080/03610918.2019.1704418 | Zbl:1489.62372
  • Colmenares, Pedro J. Work and heat distributions of an inertial Brownian particle, Physical Review E, Volume 105 (2022) no. 4 | DOI:10.1103/physreve.105.044109
  • Zheng, Chunming; Tönjes, Ralf Noise-induced swarming of active particles, Physical Review E, Volume 106 (2022) no. 6 | DOI:10.1103/physreve.106.064601
  • Chevalier, Guillaume; Coqueret, Guillaume; Raffinot, Thomas Supervised portfolios, Quantitative Finance, Volume 22 (2022) no. 12, pp. 2275-2295 | DOI:10.1080/14697688.2022.2122543 | Zbl:1518.91239
  • Gaunt, Robert E. The basic distributional theory for the product of zero mean correlated normal random variables, Statistica Neerlandica, Volume 76 (2022) no. 4, pp. 450-470 | DOI:10.1111/stan.12267 | Zbl:1528.60017
  • Pan, Yajuan; Jiang, Hui Asymptotic properties for quadratic functionals of linear self-repelling diffusion process and applications, Stochastic Analysis and Applications, Volume 40 (2022) no. 4, pp. 691-713 | DOI:10.1080/07362994.2021.1950013 | Zbl:1492.60068
  • Qu, Zhijie; Lindley, Ryan; Bregman, Joel N. The Warm Gas in the Milky Way: The Kinematical Model of C iv and Its Connection to Si iv, The Astrophysical Journal, Volume 924 (2022) no. 2, p. 86 | DOI:10.3847/1538-4357/ac35cd
  • Heyes, D. M.; Dini, D. Intrinsic viscuit probability distribution functions for transport coefficients of liquids and solids, The Journal of Chemical Physics, Volume 156 (2022) no. 12 | DOI:10.1063/5.0083228
  • Gaunt, Robert E. Stein's method and the distribution of the product of zero mean correlated normal random variables, Communications in Statistics. Theory and Methods, Volume 50 (2021) no. 2, pp. 280-285 | DOI:10.1080/03610926.2019.1634210 | Zbl:7532120
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  • García-Pérez, Alfonso New Robust Cross-Variogram Estimators and Approximations of Their Distributions Based on Saddlepoint Techniques, Mathematics, Volume 9 (2021) no. 7, p. 762 | DOI:10.3390/math9070762
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  • Li, Yang; He, Qian; Blum, Rick S. On the Product of Two Correlated Complex Gaussian Random Variables, IEEE Signal Processing Letters, Volume 27 (2020), p. 16 | DOI:10.1109/lsp.2019.2953634
  • Seijas-Macias, Antonio; Oliveira, Amílcar; Oliveira, Teresa, INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2019, Volume 2293 (2020), p. 420103 | DOI:10.1063/5.0026716
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  • Thyagarajan, Nithyanandan; Carilli, Chris L.; Nikolic, Bojan; Kent, James; Mesinger, Andrei; Kern, Nicholas S.; Bernardi, Gianni; Matika, Siyanda; Abdurashidova, Zara; Aguirre, James E.; Alexander, Paul; Ali, Zaki S.; Balfour, Yanga; Beardsley, Adam P.; Billings, Tashalee S.; Bowman, Judd D.; Bradley, Richard F.; Burba, Jacob; Carey, Steve; Cheng, Carina; DeBoer, David R.; Dexter, Matt; Acedo, Eloy de Lera; Dillon, Joshua S.; Ely, John; Ewall-Wice, Aaron; Fagnoni, Nicolas; Fritz, Randall; Furlanetto, Steven R.; Gale-Sides, Kingsley; Glendenning, Brian; Gorthi, Deepthi; Greig, Bradley; Grobbelaar, Jasper; Halday, Ziyaad; Hazelton, Bryna J.; Hewitt, Jacqueline N.; Hickish, Jack; Jacobs, Daniel C.; Julius, Austin; Kerrigan, Joshua; Kittiwisit, Piyanat; Kohn, Saul A.; Kolopanis, Matthew; Lanman, Adam; La Plante, Paul; Lekalake, Telalo; Lewis, David; Liu, Adrian; MacMahon, David; Malan, Lourence; Malgas, Cresshim; Maree, Matthys; Martinot, Zachary E.; Matsetela, Eunice; Molewa, Mathakane; Morales, Miguel F.; Mosiane, Tshegofalang; Neben, Abraham R.; Parsons, Aaron R.; Patra, Nipanjana; Pieterse, Samantha; Pober, Jonathan C.; Razavi-Ghods, Nima; Ringuette, Jon; Robnett, James; Rosie, Kathryn; Sims, Peter; Smith, Craig; Syce, Angelo; Williams, Peter K. G.; Zheng, Haoxuan Detection of cosmic structures using the bispectrum phase. II. First results from application to cosmic reionization using the Hydrogen Epoch of Reionization Array, Physical Review D, Volume 102 (2020) no. 2 | DOI:10.1103/physrevd.102.022002
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  • Seijas-Macías, Antonio; Oliveira, Amílcar; Oliveira, Teresa A.; Leiva, Víctor Approximating the Distribution of the Product of Two Normally Distributed Random Variables, Symmetry, Volume 12 (2020) no. 8, p. 1201 | DOI:10.3390/sym12081201
  • Watts, D. J.; Addison, G. E.; Bennett, C. L.; Weiland, J. L. Beyond Optical Depth: Future Determination of Ionization History from the Cosmic Microwave Background, The Astrophysical Journal, Volume 889 (2020) no. 2, p. 130 | DOI:10.3847/1538-4357/ab5fd5
  • Heyes, D. M.; Dini, D.; Smith, E. R. Single trajectory transport coefficients and the energy landscape by molecular dynamics simulations, The Journal of Chemical Physics, Volume 152 (2020) no. 19 | DOI:10.1063/5.0005600
  • Wilson, D. Keith; Ostashev, Vladimir E.; Pettit, Chris L. Distribution of the two-point product of complex amplitudes in the fully saturated scattering regime, The Journal of the Acoustical Society of America, Volume 148 (2020) no. 4, p. EL347 | DOI:10.1121/10.0002143
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  • Gaunt, Robert E. A note on the distribution of the product of zero-mean correlated normal random variables, Statistica Neerlandica, Volume 73 (2019) no. 2, pp. 176-179 | DOI:10.1111/stan.12152 | Zbl:7788772
  • Cederbaum, Jona; Scheipl, Fabian; Greven, Sonja Fast symmetric additive covariance smoothing, Computational Statistics and Data Analysis, Volume 120 (2018), pp. 25-41 | DOI:10.1016/j.csda.2017.11.002 | Zbl:1469.62031
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  • Soares Koshiyama, Adriano; Firoozye, Nick Avoiding Backtesting Overfitting by Covariance-Penalties: An Empirical Investigation of the Ordinary and Total Least Squares Cases, SSRN Electronic Journal (2018) | DOI:10.2139/ssrn.3385636
  • Cui, Guolong; Yu, Xianxiang; Iommelli, Salvatore; Kong, Lingjiang Exact Distribution for the Product of Two Correlated Gaussian Random Variables, IEEE Signal Processing Letters, Volume 23 (2016) no. 11, p. 1662 | DOI:10.1109/lsp.2016.2614539

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