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Thomas Royen | |
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Born | Thomas Royen 6 July 1947 |

Residence | Schwalbach am Taunus, Germany |

Citizenship | Germany |

Alma mater | Goethe University Frankfurt University of Freiburg Technical University Dortmund (PhD) |

Known for | Proof of Gaussian correlation inequality |

Scientific career | |

Fields | Mathematics, Statistics |

Thesis | On Convergence Against Stable Laws (1975) |

**Thomas Royen** (born July 6, 1947 in Frankfurt am Main) is a retired German professor of statistics who has been affiliated with the University of Applied Sciences Bingen. Royen came to prominence in the spring of 2017 for a relatively simple proof for the Gaussian Correlation Inequality (GCI), a conjecture that originated in the 1950s, which he had published three years earlier without much recognition.^{ [1] } A proof of this conjecture, which lies at the intersection of geometry, probability theory and statistics, had eluded top experts for decades.^{ [2] }

Royen was born in 1947 to Paul Royen, a professor with the institute for inorganic chemistry at the Goethe University Frankfurt and Elisabeth Royen, also a chemist. From 1966 to 1971, he studied mathematics and physics at his father's university and the University of Freiburg. After graduating, he worked as a tutor at the University of Freiburg, before transferring to the Technical University of Dortmund for his doctoral thesis. After attaining his PhD in 1975 with a thesis called *Über die Konvergenz gegen stabile Gesetze* (*On Convergence Against Stable Laws*), he worked as a wissenschaftlicher Assistent at Dortmund University's institute for statistics. Married with children, Royen lives in Schwalbach am Taunus.

In 1977, Royen started working as a statistician for the pharmaceutical company Hoechst AG. From 1979 to 1985, he worked at the company's own educational facility teaching mathematics and statistics. Starting in 1985 until becoming an emeritus in 2010, he taught statistics and mathematics at the University of Applied Sciences Bingen in Rhineland-Palatinate.^{ [3] }

Royen worked mainly on probability distributions, in particular multivariate chi-squares and gamma distributions, to improve some frequently used statistical test procedures. Nearly half of his circa 30 publications were written when he was aged over sixty. Because he was annoyed over some contradictory reviews and in a few cases also over the incompetence of a referee, he decided in his later years, when his actions had no influence anymore on his further career, to publish his papers on the online platform arXiv.org and sometimes in a less renowned Indian journal to fulfill, at least formally, the condition of a peer review.^{ [4] }

On July 17, 2014, a few years after his retirement, when brushing his teeth, Royen had a flash of insight: how to use the Laplace transform of the multivariate gamma distribution to achieve a relatively simple proof for the Gaussian correlation inequality, a conjecture on the intersection of geometry, probability theory and statistics, formulated after work by Dunnett and Sobel (1955) and the American statistician Olive Jean Dunn (1958),^{ [5] } that had remained unsolved since then. He sent a copy of his proof to Donald Richards, an acquainted American mathematician, who worked on a proof of the GCI for 30 years. Richards immediately saw the validity of Royen's proof and subsequently helped him to transform the mathematical formulas into LaTeX. When Royen contacted other reputed mathematicians, though, they didn't bother to investigate his proof, because Royen was relatively unknown, and these mathematicians therefore estimated the chance that Royen's proof would be false as very high.^{ [2] }

Royen published this proof in an article with the title *A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions* on arXiv ^{ [6] } and subsequently in the *Far East Journal of Theoretical Statistics*,^{ [7] } a relatively unknown periodical based in Allahabad, India, for which Royen was at the time voluntarily working as a referee himself. Due to this, his proof went at first largely unnoticed by the scientific community,^{ [8] } until in late 2015 two Polish mathematicians, Rafał Latała and Dariusz Matlak, wrote a paper in which they reorganized Royen's proof in a way that was intended to be easier to follow.^{ [1] } In July 2015, Royen supplemented his proof with a further paper in arXiv *Some probability inequalities for multivariate gamma and normal distributions*.^{ [9] }

On March 28, 2017 Natalie Wolchover of * Quanta Magazine * published a story about Royen's proof, after which he gained more academic and public recognition for his achievement, which is not only remarkable because of its mathematical significance but also of Royen's (from a mathematical perspective) advanced age (67) when he found the proof of the GCI. In addition to this, it's also noteworthy that the world of mathematics and statistics has apparently become so vast that experts in a field can be unaware two years after publication of a proof, that an important conjecture in their own field has been proved.

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The **Gaussian correlation inequality** (**GCI**), formerly known as the **Gaussian correlation conjecture** (**GCC**), is a mathematical theorem in the fields of mathematical statistics and convex geometry. A special case of the inequality was published as a conjecture in a paper from 1955; further development was given by Olive Jean Dunn in 1958. The general case was stated in 1972, also as a conjecture.

- 1 2 Rafal Latala and Dariusz Matlak, Royen's proof of the Gaussian correlation inequality, arXiv:1512.08776
- 1 2 "A Long-Sought Proof, Found and Almost Lost".
*Quanta Magazine*. Natalie Wolchover. March 28, 2017. Retrieved May 1, 2017. - ↑ "Curriculum Vitae Thomas Royen".
*Forschungsplattform des Landes Rheinland-Pfalz*. Landesregierung Rheinland-Pfalz. - ↑ "Der Beweis" (in German). Sibylle Anderl. April 7, 2017.
- ↑ Dunn, Olive Jean (March 1959). "Estimation of the medians for dependent variables".
*The Annals of Mathematical Statistics*.**30**(1): 192–197. doi:10.1214/aoms/1177706374. JSTOR 2237135. - ↑ Royen, T. (2014). "A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions". arXiv: 1408.1028 . Supplemented by Royen, Thomas (2015). "Some probability inequalities for multivariate gamma and normal distributions". arXiv: 1507.00528 .
- ↑ Thomas Royen:
*A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions,*in:*Far East Journal of Theoretical Statistics*, Part 48 Nr. 2, Pushpa Publishing House, Allahabad 2014, p.139–145 - ↑ In the Quanta magazine article for instance Tilmann Gneiting, a statistician at the Heidelberg Institute for Theoretical Studies, just 65 miles from Bingen, said he was shocked to learn in July 2016, two years after the fact, that the GCI had been proved.
- ↑ Royen, Thomas (2015-07-02). "Some probability inequalities for multivariate gamma and normal distributions". arXiv: 1507.00528 [math.PR].

- Profile of Thomas Royen on Science Portal Rheinland-Pfalz (RLP)
- Thomas Royen, "A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions", arXiv : 1408.1028
- Thomas Royen, "Some probability inequalities for multivariate gamma and normal distributions", arXiv : 1507.00528
- Thomas Royen, "A note on the existence of the multivariate gamma distribution", arXiv : 1606.04747
- (in French) Bourbaki, Frank Barthe on YouTube. January 14, 2017
- Informal interview with Thomas Royen at University of Applied Sciences Bingen on YouTube (in German)

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