A note on the distribution of the product of zero‐mean correlated normal random variables
The problem of finding an explicit formula for the probability density function of two zero‐mean correlated normal random variables dates back to 1936. Perhaps, surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which...
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| Published in | Statistica Neerlandica Vol. 73; no. 2; pp. 176 - 179 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Oxford
Blackwell Publishing Ltd
01.05.2019
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| Subjects | |
| Online Access | Get full text |
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| Abstract | The problem of finding an explicit formula for the probability density function of two zero‐mean correlated normal random variables dates back to 1936. Perhaps, surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero‐mean correlated normal random variables as a variance‐gamma random variable, from which an explicit formula for the probability density function is immediate. |
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| AbstractList | The problem of finding an explicit formula for the probability density function of two zero‐mean correlated normal random variables dates back to 1936. Perhaps, surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero‐mean correlated normal random variables as a variance‐gamma random variable, from which an explicit formula for the probability density function is immediate. |
| Author | Gaunt, Robert E. |
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| Cites_doi | 10.1007/978-3-0348-7943-9_15 10.4324/9780203809556 10.3758/BF03193007 10.1103/PhysRevE.77.036318 10.1086/296519 10.1214/aoms/1177732541 10.1109/LSP.2016.2614539 10.1142/S0219024908005093 10.1007/978-1-4612-0173-1 10.1016/B978-044450896-6.50008-X |
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| References_xml | – start-page: 211 year: 2003 end-page: 248 – volume: 39 start-page: 384 year: 2007 end-page: 389 article-title: Distribution of the product confidence limits for the indirect effect: program PRODCLIN publication-title: Behavior Research Methods – volume: 63 start-page: 511 year: 1990 end-page: 524 article-title: The variance gamma (V.G.) model for share market returns publication-title: The Journal of Business – volume: 19 start-page: 1 issue: 38 year: 2014 end-page: 33 article-title: Variance‐gamma approximation via Stein's method publication-title: Electronic Journal of Probability – volume: 58 start-page: 105 year: 2004 end-page: 153 – volume: 354 start-page: 201 year: 2016 end-page: 204 article-title: On the distribution of the product of correlated normal random variables publication-title: Comptes Rendus de l'Académie des Sciences ‐ Series I – year: 2007 – year: 2001 – volume: 7 start-page: 1 year: 1936 end-page: 15 article-title: On the frequency function of . publication-title: Annals of Mathematical Statistics – volume: 23 start-page: 1662 year: 2016 end-page: 1666 article-title: Exact distribution for the product of two correlated Gaussian random variables publication-title: IEEE Signal Processing Letters – volume: 77 year: 2008 article-title: Craig's distribution and the statistics of Lagrangian power in two‐dimensional turbulence publication-title: Physical Review E – volume: 11 start-page: 943 year: 2008 end-page: 955 article-title: Option pricing with VG‐like models publication-title: International Journal of Theoretical and Applied Finance – year: 2013 – year: 2012 – ident: e_1_2_5_6_1 doi: 10.1007/978-3-0348-7943-9_15 – ident: e_1_2_5_16_1 – ident: e_1_2_5_12_1 doi: 10.4324/9780203809556 – ident: e_1_2_5_13_1 doi: 10.3758/BF03193007 – ident: e_1_2_5_2_1 doi: 10.1103/PhysRevE.77.036318 – volume: 354 start-page: 201 year: 2016 ident: e_1_2_5_15_1 article-title: On the distribution of the product of correlated normal random variables publication-title: Comptes Rendus de l'Académie des Sciences ‐ Series I – ident: e_1_2_5_8_1 – volume-title: Table of integrals, series, and products year: 2007 ident: e_1_2_5_10_1 – ident: e_1_2_5_14_1 doi: 10.1086/296519 – ident: e_1_2_5_4_1 doi: 10.1214/aoms/1177732541 – ident: e_1_2_5_5_1 doi: 10.1109/LSP.2016.2614539 – ident: e_1_2_5_7_1 doi: 10.1142/S0219024908005093 – ident: e_1_2_5_11_1 doi: 10.1007/978-1-4612-0173-1 – volume: 19 start-page: 1 issue: 38 year: 2014 ident: e_1_2_5_9_1 article-title: Variance‐gamma approximation via Stein's method publication-title: Electronic Journal of Probability – ident: e_1_2_5_3_1 doi: 10.1016/B978-044450896-6.50008-X |
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| SubjectTerms | Correlation Mathematical analysis Monte Carlo simulation probability density function Probability density functions product of correlated normal random variables Random variables variance‐gamma distribution |
| Title | A note on the distribution of the product of zero‐mean correlated normal random variables |
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