Paul Pierre Lévy
Encyclopedia
Paul Pierre Lévy was a Jewish French
mathematician who was active especially in probability theory
, introducing martingale
s and Lévy flight
s. Lévy process
es, Lévy measures, Lévy's constant
, the Lévy distribution, the Lévy skew alpha-stable distribution, the Lévy area, the Lévy arcsine law
, and the fractal
Lévy C curve
are also named after him.
Lévy was born in Paris
, the son of Lucien Lévy, an Examiner at the École Polytechnique
. Lévy also attended the École Polytechnique and published his first paper in 1905 at the age of 19, while still an undergraduate. 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.
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
Statute on Jews
.
Lévy received a number of honors, including membership at the French Academy of Sciences
and honorary membership at the London Mathematical Society
.
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician who was active especially in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, introducing martingale
Martingale (probability theory)
In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...
s and Lévy flight
Lévy flight
A Lévy flight is a random walk in which the step-lengths have a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions...
s. Lévy process
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...
es, Lévy measures, Lévy's constant
Lévy's constant
In mathematics Lévy's constant occurs in an expression for the asymptotic behaviour of the denominators of the convergents of continued fractions....
, the Lévy distribution, the Lévy skew alpha-stable distribution, the Lévy area, the Lévy arcsine law
Lévy arcsine law
In probability theory, the Lévy arcsine law, found by , states that the probability distribution of the proportion of the time that a Wiener process is positive is a random variable whose probability distribution is the arcsine distribution...
, and the fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
Lévy C curve
Lévy C curve
In mathematics, the Lévy C curve is a self-similar fractal that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and G...
are also named after him.
Lévy was born in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
, the son of Lucien Lévy, an Examiner at the École Polytechnique
École Polytechnique
The École Polytechnique is a state-run institution of higher education and research in Palaiseau, Essonne, France, near Paris. Polytechnique is renowned for its four year undergraduate/graduate Master's program...
. Lévy also attended the École Polytechnique and published his first paper in 1905 at the age of 19, while still an undergraduate. His teacher and advisor was Jacques Hadamard
Jacques Hadamard
Jacques Salomon Hadamard FRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...
. 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.
During World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
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
Benoît Mandelbrot
Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...
and Georges Matheron
Georges Matheron
Georges François Paul Marie Matheron was a French mathematician and geologist, known as the founder of geostatistics and a co-founder of mathematical morphology. In 1968 he created the Centre de Géostatistique et de Morphologie Mathématique at the Paris School of Mines in Fontainebleau...
. 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
Vichy France
Vichy France, Vichy Regime, or Vichy Government, are common terms used to describe the government of France that collaborated with the Axis powers from July 1940 to August 1944. This government succeeded the Third Republic and preceded the Provisional Government of the French Republic...
Statute on Jews
Statute on Jews
The Statute on Jews was discriminatory legislation against French Jews passed on October 3, 1940 by the Vichy Regime, grouping them as a lower class and depriving them of citizenship before rounding them up at Drancy internment camp then taking them to be exterminated in concentration camps...
.
Lévy received a number of honors, including membership at the French Academy of Sciences
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...
and honorary membership at the London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...
.
See also
- Lévy metricLévy metricIn mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Pierre Lévy.-Definition:...
- Lévy's modulus of continuityLévy's modulus of continuityIn mathematics, Lévy's modulus of continuity theorem gives a result about the almost sure behaviour of an estimate of the modulus of continuity for the Wiener process, which models Brownian motion. It is due to the French mathematician Paul Pierre Lévy....
- Lévy–Prokhorov metric
- Lévy's continuity theoremLévy's continuity theoremIn probability theory, the Lévy’s continuity theorem, named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions...
- Lévy's zero-one law
- Concentration of measureConcentration of measureIn mathematics, concentration of measure is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that "A random variable that depends in a Lipschitz way on many independent variables ...