Thomas Royen | |
---|---|

Born | Thomas Royen 6 July 1947 |

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.

In probability theory and statistics, a **Gaussian process** is a stochastic process, such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.

The **Kepler conjecture**, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.

**Probability** is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "A specific event will occur." The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

In probability theory, the central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases to infinity. Under stronger assumptions, the **Berry–Esseen theorem**, or **Berry–Esseen inequality**, gives a more quantitative result, because it also specifies the rate at which this convergence takes place by giving a bound on the maximal error of approximation between the normal distribution and the true distribution of the scaled sample mean. The approximation is measured by the Kolmogorov–Smirnov distance. In the case of independent samples, the convergence rate is *n*^{−1/2}, where *n* is the sample size, and the constant is estimated in terms of the third absolute normalized moments.

In convex analysis, a non-negative function *f* : **R**^{n} → **R**_{+} is **logarithmically concave** if its domain is a convex set, and if it satisfies the inequality

In probability theory and mathematical physics, a **random matrix** is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.

Inequalities are very important in the study of information theory. There are a number of different contexts in which these inequalities appear.

In mathematics, the **Fortuin–Kasteleyn–Ginibre (FKG) inequality** is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics, due to Cees M. Fortuin, Pieter W. Kasteleyn, and Jean Ginibre (1971). Informally, it says that in many random systems, increasing events are positively correlated, while an increasing and a decreasing event are negatively correlated. It was obtained by studying the random cluster model.

This page lists articles related to probability theory. In particular, it lists many articles corresponding to specific probability distributions. Such articles are marked here by a code of the form (X:Y), which refers to number of random variables involved and the type of the distribution. For example (2:DC) indicates a distribution with two random variables, discrete or continuous. Other codes are just abbreviations for topics. The list of codes can be found in the table of contents.

The **Tracy–Widom distribution**, introduced by Craig Tracy and Harold Widom, is the probability distribution of the normalized largest eigenvalue of a random Hermitian matrix.

**Gábor J. Székely** is a Hungarian-American statistician/mathematician best known for introducing the Energy of data [see E-statistics or *Package energy* in R ], e.g. the distance correlation which is a bona fide dependence measure, equals zero exactly when the variables are independent, the distance skewness which equals zero exactly when the probability distribution is diagonally symmetric, the E-statistic for normality test and the E-statistic for clustering.

**Inter-universal Teichmüller theory** is the name given by mathematician Shinichi Mochizuki to a theory he developed in the 2000s, following his earlier work in arithmetic geometry. According to Mochizuki, it is "an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve". The theory was made public in a series of four preprints posted in 2012 to his website. The most striking claimed application of the theory is to provide a proof for various outstanding conjectures in number theory, in particular the abc conjecture. Mochizuki and a few other mathematicians claim that the theory indeed yields such a proof but this has so far not been accepted by the mathematical community.

**Julius Bogdan Borcea** was a Romanian Swedish mathematician. His scientific work included vertex operator algebra and zero distribution of polynomials and entire functions, via correlation inequalities and statistical mechanics.

**Olive Jean Dunn** was an American mathematician and statistician, and professor of biostatistics at the University of California Los Angeles (UCLA). She described methods for computing confidence intervals and also codified the Bonferroni correction's application to confidence intervals. She authored the textbook *Basic Statistics: A Primer for the Biomedical Sciences* in 1977.

**Xinyi Yuan** is a Chinese mathematician who is currently a professor of mathematics at Peking University working in number theory, arithmetic geometry, and automorphic forms. In particular, his work focuses on arithmetic intersection theory, algebraic dynamics, Diophantine equations and special values of L-functions.

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.

**Friedrich Götze** is a German mathematician, specializing in probability theory, mathematical statistics, and number theory.

- 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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