267:=2, one can think of darts being thrown at a board, with their landing spots in the plane distributed according to a 2-variable normal distribution centered at the origin. (This is a reasonable assumption for any given darts player, with different players being described by different normal distributions.) If we now consider a circle and a rectangle in the plane, both centered at the origin, then the proportion of the darts landing in the intersection of both shapes is no less than the product of the proportions of the darts landing in each shape. This can also be formulated in terms of
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The gaussian correlation inequality states that probability of hitting both circle and rectangle with a dart is greater than or equal to the product of the individual probabilities of hitting the circle or the
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439:. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Vol. II: Probability theory. Berkeley, California: Univ. California Press. pp. 241β265.
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The conjecture, and its solution, came to public attention in 2017, when other mathematicians described Royen's proof in a mainstream publication and popular media reported on the story.
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Dunnett, C. W.; Sobel, M. (1955). "Approximations to the probability integral and certain percentage points of a multivariate analogue of
Student's t -distribution".
302:. Another reason was a history of false proofs (by others) and many failed attempts to prove the conjecture, causing skepticism among mathematicians in the field.
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271:: if you're informed that your last dart hit the rectangle, then this information will increase your estimate of the probability that the dart hit the circle.
294:, a retired German statistician, proved it using relatively elementary tools. In fact, Royen generalized the conjecture and proved it for multivariate
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Inequalities on the probability content of convex regions for elliptically contoured distributions
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in 1958. The general case was stated in 1972, also as a conjecture. The case of dimension
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LataΕa, R.; Matlak, D. (2017). "Royen's Proof of the
Gaussian Correlation Inequality".
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Das Gupta, S.; Eaton, M. L.; Olkin, I.; Perlman, M.; Savage, L. J.; Sobel, M. (1972).
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485:"A Particular Case of Correlation Inequality for the Gaussian Measure"
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The general case of the inequality remained open until 2014, when
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452:"A Gaussian Correlation Inequality for Symmetric Convex Sets"
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Schechtman, G.; Schlumprecht, Th.; Zinn, J. (January 1998).
253:{\displaystyle \mu (E\cap F)\geq \mu (E)\cdot \mu (F).}
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110:{\displaystyle \mathbb {R} ^{n}}
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492:The Annals of Probability
456:The Annals of Probability
392:The Annals of Probability
335:10.1093/biomet/42.1-2.258
269:conditional probabilities
33:), formerly known as the
263:As a simple example for
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47:mathematical statistics
709:Geometric inequalities
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