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Paul Lévy (mathematician)

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Paul Lévy
Paul Pierre Lévy
Born(1886-09-15)15 September 1886
Died15 December 1971(1971-12-15) (aged 85)
Alma materUniversity of Paris
Known forAdditive process
Brownian excursion
Concentration of measure
Martingale (probability theory)
Universal chord theorem
Lévy alpha-stable distribution
Lévy's arcsine law
Lévy C curve
Lévy's constant
Lévy characterisation
Lévy's continuity theorem
Lévy distribution
Lévy flight
Lévy's local time
Lévy measure
Lévy's modulus of continuity theorem
Lévy process
Lévy's zero–one law
Lévy–Khintchine representation
Lévy–Prokhorov metric
Lévy–Steinitz theorem
Lindeberg–Lévy CLT
Wiener–Lévy theorem
AwardsEmile Picard Medal of the French Academy of Sciences (1953)[1]
Scientific career
InstitutionsÉcole Polytechnique
École des Mines
Doctoral advisorJacques Hadamard
Vito Volterra
Doctoral studentsWolfgang Doeblin
Michel Loève
Benoît Mandelbrot
Georges Matheron

Paul Pierre Lévy (15 September 1886 – 15 December 1971)[2] was a French mathematician who was active especially in probability theory, introducing fundamental concepts such as local time, stable distributions and characteristic functions. Lévy processes, Lévy flights, Lévy measures, Lévy's constant, the Lévy distribution, the Lévy area, the Lévy arcsine law, and the fractal Lévy C curve are named after him.


Lévy was born in Paris to a Jewish family which already included several mathematicians.[3] His father Lucien Lévy was an examiner at the École Polytechnique. Lévy attended the École Polytechnique and published his first paper in 1905, at the age of nineteen, while still an undergraduate, in which he introduced the Lévy–Steinitz theorem. His teacher and advisor was Jacques Hadamard. After graduation, he spent a year in military service and then studied for three years at the École des Mines, where he became a professor in 1913.[2]

During World War I Lévy conducted mathematical analysis work for the French Artillery. In 1920 he was appointed Professor of Analysis at the École Polytechnique, where his students included Benoît Mandelbrot and Georges Matheron. He remained at the École Polytechnique until his retirement in 1959, with a gap during World War II after his 1940 firing because of the Vichy anti-Jewish legislation.[2]

Lévy made many fundamental contributions to probability theory and the nascent theory of stochastic processes. He introduced the notion of 'stable distribution' which share the property of stability under addition of independent variables and proved a general version of the Central Limit theorem, recorded in his 1937 book Théorie de l'addition des variables aléatoires, using the notion of characteristic function. He also introduced, independently from Aleksandr Khinchin, the notion of infinitely divisible law and derived their characterization through the Lévy–Khintchine representation.

His 1948 monograph on Brownian motion, Processus stochastiques et mouvement brownien, contains a wealth of new concepts and results, including the Lévy area, the Lévy arcsine law, the local time of a Brownian path, and many other results.

Lévy received a number of honours, including membership at the French Academy of Sciences and honorary membership at the London Mathematical Society.

His daughter Marie-Hélène Schwartz and son-in-law Laurent Schwartz were also notable mathematicians.[4]


  • 1922 – Lecons d'analyse Fonctionnelle
  • 1925 – Calcul des probabilités
  • 1937 – Théorie de l'addition des variables aléatoires
  • 1948 – Processus stochastiques et mouvement brownien
  • 1954 – Le mouvement brownien

See also[edit]


  1. ^ Médaille Emile Picard
  2. ^ a b c O'Connor, John J.; Robertson, Edmund F., "Paul Lévy (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews
  3. ^ Barbut, Marc; Locker, Bernard; Mazliak, Laurent (2013). Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters. p. xii. ISBN 978-1-4471-5618-5.
  4. ^ Kosmann-Schwarzbach, Yvette (2015), "Women mathematicians in France in the mid-twentieth century", BSHM Bulletin: Journal of the British Society for the History of Mathematics, 30 (3): 227–242, arXiv:1502.07597, doi:10.1080/17498430.2014.976804, S2CID 119148294.

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