Paul Lévy

How to use this course

Paul Lévy is one of the founders of modern probability theory. Many objects every probabilist uses carry his name: Lévy processes, the Lévy–Khintchine formula, Lévy’s continuity theorem, Lévy’s characterization of Brownian motion, Lévy’s arcsine laws, Lévy’s local time. He worked mostly alone, in a style driven by intuition about paths and sums, and he was often ahead of the rigorous foundations of his day. This course follows his life and then rebuilds his main results in modern notation.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The order follows his career: analysis (1905–1930s), sums of independent variables (1919–1937), martingales (1934–1935) and Brownian motion (1937–1948).

Timeline

A family of mathematicians, 1886–1907

Paul Pierre Lévy was born in Paris on 15 September 1886 into a Jewish family with a mathematical tradition. His paternal grandfather, Édouard Lévy, had studied at the École Normale Supérieure and taught mathematics. His father, Lucien Lévy, was a graduate of the École Polytechnique, taught at Paris lycées including Louis-le-Grand, examined candidates for the Polytechnique, and served as president of the Société Mathématique de France in 1911. His mother was Alice Flora Wolff, whose father was an inspector general of the Ponts et Chaussées.

Lévy attended the Lycée Montaigne, then Louis-le-Grand, and finally the Lycée Saint-Louis, where he prepared for the competitive entrance examinations of the grandes écoles. He placed first for the École Normale Supérieure and second for the École Polytechnique. He chose the Polytechnique, which trained engineers and state officials, very likely because of his father’s connection to it.

A first theorem at nineteen

While still a student he published Sur les séries semi-convergentes (1905). Riemann had shown that a conditionally convergent real series can be rearranged to have any sum. Lévy asked what happens for series of vectors in the plane or in space, and found that the possible sums form a line, a plane or the whole space. His argument had a gap in higher dimensions, which Ernst Steinitz filled in 1913; the result is now the Lévy–Steinitz theorem (Lesson 04). A second early paper dealt with the density of prime numbers.

He graduated first in his class, served a year in the artillery in 1906–07, and then entered the École des Mines, the school of the elite Corps des Mines.

Hadamard and functional analysis, 1907–1914

At the École des Mines Lévy trained as an engineer, but he also followed courses at the Sorbonne and at the Collège de France. The decisive teacher was Jacques Hadamard. In his 1910 course at the Collège de France, Hadamard discussed functionals: quantities that depend on a whole curve or function, which Vito Volterra had called “functions of lines”. Lévy took his thesis subject from that course.

The thesis, Sur les équations intégro-différentielles définissant des fonctions de lignes, studied equations in which a functional varies as its argument curve is deformed. It is an example of Hadamard’s programme of extending the calculus to infinitely many variables. Lévy defended it in 1911 before a jury of Émile Picard, Henri Poincaré and Hadamard. The Mathematics Genealogy Project lists Hadamard and Volterra as his advisors; Hadamard was the lasting influence.

In 1913 Lévy became professor of mathematics at the École des Mines in Paris, a post he held until 1951, and began work as a tutor at the Polytechnique. The same year he married Suzanne Lévy. They had three children: Marie-Hélène (born 1913), Denise (1916) and Jean-Claude (1918). Marie-Hélène became a mathematician and married Laurent Schwartz.

War, and the turn of 1919

During the First World War Lévy served in the artillery. He was assigned mathematical problems of ballistics and of defence against air attack. René Gateaux, a young analyst working on the same questions of functional calculus, was killed in 1914. After the war Hadamard asked Lévy to prepare Gateaux’s unpublished papers, and Lévy published them in 1919 with his own additions. Gateaux’s work on differentials of functionals, now the Gateaux derivative, reached readers through this edition.

Three lectures

Lévy’s entry into probability was an accident of timetabling. In 1919 Georges Humbert, the professor of analysis at the Polytechnique, was ill, and Lévy was asked to give three lectures on the calculus of probabilities and the role of the Gaussian law in the theory of errors. Preparing them, he asked why the Gaussian law should be the universal limit of sums of small errors. He looked for all laws that are stable under addition, and found a whole family of them (Lesson 06).

Probability in France was then a marginal subject, associated with gambling and actuarial work, and treated with suspicion by many pure mathematicians. Lévy had no teacher in it. He worked out its basic tools, above all the characteristic function, for himself. In 1920 he succeeded Humbert as professor of analysis at the Polytechnique. His Calcul des probabilités appeared in 1925 and his Leçons d’analyse fonctionnelle in 1922. For the next fifty years he would publish mostly on probability.

Analysis: rearrangements, chords, concentration

For the general reader. Before and alongside probability, Lévy proved theorems in classical and functional analysis that are still taught. Three are gathered here: what happens when you reorder an infinite sum of vectors, a surprising fact about horizontal chords of a graph, and the discovery that on a high-dimensional sphere almost all the area lies close to every equator.

Rearranging vector series

For \(d=1\) this is Riemann’s theorem: either the series converges absolutely (\(\Gamma=\{0\}\)) or every real number is a sum. The hard step in higher dimensions is the Steinitz lemma: if vectors \(u_1,\dots,u_m\) of norm at most 1 sum to 0, they can be reordered so that every partial sum has norm at most a constant \(C_d\) depending only on the dimension.

The universal chord theorem

Proof. Let \(g(x)=f(x+\tfrac1n)-f(x)\). Then \(\sum_{k=0}^{n-1}g(k/n)=f(1)-f(0)=0\), so \(g\) takes values of both signs or vanishes at a grid point; the intermediate value theorem gives a zero. Sharpness is Exercise 4.2.

Concentration of measure

In the Leçons d’analyse fonctionnelle (1922), and again in his 1951 book on functional analysis, Lévy studied functions of infinitely many variables as limits of functions on spheres of growing dimension. He stated a spherical isoperimetric inequality: among all subsets of the sphere with a given area, a spherical cap has the smallest \(\varepsilon\)-neighbourhood. Erhard Schmidt gave a rigorous proof in 1948. Vitali Milman revived the idea in the 1970s as the concentration of measure phenomenon.

The bound follows from the isoperimetric inequality: the set \(A=\{f\le m_f\}\) has measure at least \(\tfrac12\), so its \(r\)-neighbourhood is at least as large as that of a hemisphere, and the complement of the \(r\)-neighbourhood of a hemisphere is a cap of measure at most \(e^{-(n-1)r^2/2}\). A Lipschitz function on a high-dimensional sphere is thus almost constant.

Lévy also extended Norbert Wiener’s lemma on Fourier series. The Wiener–Lévy theorem (1935) says: if \(f\) has an absolutely convergent Fourier series and \(F\) is analytic on a neighbourhood of the range of \(f\), then \(F\circ f\) also has an absolutely convergent Fourier series.

Characteristic functions and the central limit theorem

For the general reader. Adding independent random quantities is hard to do directly, because their distributions combine through convolution. Lévy made systematic use of a transform that turns this into multiplication: the characteristic function, a Fourier transform of the distribution. With it he proved the central limit theorem in its standard textbook form.

The characteristic function of a real random variable \(X\) is \(\varphi_X(u)=\mathbb E\,e^{iuX}\), \(u\in\mathbb R\). It always exists, satisfies \(|\varphi_X|\le1\) and \(\varphi_X(0)=1\), is uniformly continuous, and determines the law of \(X\). If \(X\) and \(Y\) are independent, \(\varphi_{X+Y}=\varphi_X\varphi_Y\). Laplace and Cauchy had used such transforms, and Aleksandr Lyapunov used them in his 1901 proof of the central limit theorem. Lévy made them the central tool and proved the theorem that justifies their use.

Proof sketch of 2. The key estimate links the tails of a law to its characteristic function near 0: for \(\delta>0\),

By dominated convergence the right side tends to \(\frac1\delta\int_{-\delta}^{\delta}(1-\varphi(u))\,du\), which is small for small \(\delta\) by continuity of \(\varphi\) at 0. So the sequence is tight. By Helly’s selection theorem every subsequence has a further subsequence converging in distribution, and by part 1 the limit has characteristic function \(\varphi\). Since \(\varphi\) determines the law, all subsequential limits agree.

Proof. Take \(\mu=0,\ \sigma=1\). Then \(\varphi(u)=1-\tfrac{u^2}{2}+o(u^2)\) as \(u\to0\), so

the characteristic function of \(N(0,1)\). The continuity theorem finishes the proof. Lévy also found necessary and sufficient conditions for convergence to the normal law for non-identical summands, at the same time as William Feller (1935).

Stable laws

For the general reader. The sum of two independent Gaussian errors is again Gaussian, only wider. Lévy asked which other laws share this property. He found a whole family, indexed by a number \(\alpha\) between 0 and 2. Apart from the Gaussian (\(\alpha=2\)), they all have heavy tails: extreme values are far more likely, and the variance is infinite. These laws later became central in physics and finance.

A non-degenerate random variable \(X\) is stable if for independent copies \(X_1,\dots,X_n\) of \(X\) there are constants \(c_n>0\), \(d_n\) with

It is strictly stable if \(d_n=0\). Lévy showed that necessarily \(c_n=n^{1/\alpha}\) for some \(\alpha\in(0,2]\), the index. He then found all the characteristic functions.

The symmetric case \(\beta=\mu=0\) is the simplest: \(\varphi(u)=e^{-\sigma^\alpha|u|^\alpha}\). Then \(\varphi(u)^n=\varphi(n^{1/\alpha}u)\), which is exactly strict stability with \(c_n=n^{1/\alpha}\). That \(e^{-|u|^\alpha}\) is a characteristic function only for \(\alpha\le2\) is a non-trivial fact: for \(\alpha>2\) its second derivative at 0 vanishes, which would force zero variance.

Tails and domains of attraction

For \(0<\alpha<2\), a stable law has \(P(|X|>x)\sim Cx^{-\alpha}\) as \(x\to\infty\). Moments of order \(p\) exist exactly for \(p<\alpha\). Lévy and later Gnedenko and Doeblin showed that normalized sums \((S_n-b_n)/a_n\) of i.i.d. variables can converge only to stable laws, and described the domain of attraction of each one in terms of regularly varying tails. The Gaussian law attracts every law with finite variance; heavy-tailed laws are attracted to the other stable laws.

Three cases have closed-form densities: the normal (\(\alpha=2\)), the Cauchy law \(\frac{1}{\pi(1+x^2)}\) (\(\alpha=1,\beta=0\)), and the Lévy distribution (\(\alpha=\tfrac12,\beta=1\)) with density

Infinite divisibility and Lévy processes

For the general reader. Some random quantities can be cut into any number of independent identical pieces: a Poisson count over an hour is the sum of counts over sixty minutes. Bruno de Finetti, Andrey Kolmogorov, Lévy and Aleksandr Khinchin found the complete list of such laws between 1929 and 1937. Each one is the value at time 1 of a random process with independent, stationary increments. These processes are now called Lévy processes.

A law \(\mu\) on \(\mathbb R\) is infinitely divisible if for every \(n\) there is a law \(\mu_n\) with \(\mu=\mu_n^{*n}\). Normal, Poisson, gamma, Cauchy and all stable laws are infinitely divisible. Lévy’s 1934 paper in the Annali della Scuola Normale Superiore di Pisa found the general form by analysing processes with independent increments path by path. Khinchin gave an analytic derivation in 1937. De Finetti (1929) and Kolmogorov (1932, finite variance case) had earlier partial results.

The measure \(\nu\) is the Lévy measure: \(\nu(A)\) is the expected number of jumps with size in \(A\) per unit time. The compensating term \(-iux\mathbf 1_{\{|x|<1\}}\) makes the integral converge even when there are infinitely many small jumps.

Lévy processes and the Lévy–Itô decomposition

A Lévy process is a process \((X_t)_{t\ge0}\) with \(X_0=0\), independent and stationary increments, and paths that are right-continuous with left limits. Then \(\mathbb E\,e^{iuX_t}=e^{t\psi(u)}\) with \(\psi\) as above. Lévy saw that the process splits into independent pieces, and Kiyosi Itô (1942) made this rigorous:

where \(B\) is a Brownian motion and \(N\) an independent Poisson random measure with intensity \(ds\,\nu(dx)\). A Lévy process is a drift, plus a Brownian motion, plus big jumps, plus compensated small jumps. For the symmetric \(\alpha\)-stable process, \(\sigma=0\) and \(\nu(dx)=C|x|^{-1-\alpha}dx\).

Martingales and the zero–one law

For the general reader. A fair game is one where, whatever has happened so far, your expected gain on the next round is zero. Lévy saw that this condition, much weaker than independence, is enough to prove many limit theorems. The idea, named martingale by Jean Ville in 1939 and developed by Joseph Doob, became one of the main tools of modern probability.

Let \((\mathcal F_n)\) be a filtration, an increasing sequence of \(\sigma\)-algebras representing information available at time \(n\). An integrable adapted sequence \((M_n)\) is a martingale if \(\mathbb E[M_{n+1}\mid\mathcal F_n]=M_n\). In 1934–35 Lévy studied sums \(\sum X_k\) with \(\mathbb E[X_{k+1}\mid X_1,\dots,X_k]=0\) and showed that the central limit theorem and other laws extend to them.

Proof sketch. \(M_n=\mathbb E[X\mid\mathcal F_n]\) is a uniformly integrable martingale, so it converges a.s. and in \(L^1\) to some \(M_\infty\) by Doob’s convergence theorem. For \(A\in\mathcal F_m\) and \(n\ge m\), \(\mathbb E[M_n\mathbf 1_A]=\mathbb E[X\mathbf 1_A]\); letting \(n\to\infty\) gives \(\mathbb E[M_\infty\mathbf 1_A]=\mathbb E[X\mathbf 1_A]\) on the generating algebra \(\bigcup_m\mathcal F_m\), hence on \(\mathcal F_\infty\) by the \(\pi\)–\(\lambda\) theorem. Since \(M_\infty\) is \(\mathcal F_\infty\)-measurable, \(M_\infty=\mathbb E[X\mid\mathcal F_\infty]\). The zero–one law is the case \(X=\mathbf 1_A\).

Kolmogorov’s zero–one law follows: if \(A\) is a tail event of independent variables \(X_1,X_2,\dots\), then \(A\) is independent of each \(\mathcal F_n=\sigma(X_1,\dots,X_n)\), so \(P(A\mid\mathcal F_n)=P(A)\), and the limit \(\mathbf 1_A\) must equal the constant \(P(A)\). Lévy also proved the downward theorem: for a decreasing sequence \(\mathcal G_n\), \(\mathbb E[X\mid\mathcal G_n]\to\mathbb E[X\mid\bigcap_n\mathcal G_n]\).

Brownian motion: construction and fine structure

For the general reader. Brownian motion is the random path of a particle buffeted by molecules. Wiener had built it rigorously in 1923. Lévy gave a simple construction by repeated refinement, measured exactly how rough its paths are, and found a criterion that recognizes Brownian motion among all continuous random processes. His 1948 book on the subject was decades ahead of its time.

A standard Brownian motion is a process \((B_t)_{t\ge0}\) with \(B_0=0\), continuous paths, and independent increments with \(B_t-B_s\sim N(0,t-s)\).

Lévy’s construction

Set \(B_0=0\) and \(B_1=Z_0\sim N(0,1)\). At stage \(k\), the values at the dyadic points \(j2^{-k}\) are known. For each new midpoint \(t=(s+s')/2\) with \(s'-s=2^{-k}\), set

with fresh independent \(Z_t\sim N(0,1)\). This is the exact conditional law of Brownian motion at the midpoint given its endpoints, since the Brownian bridge over an interval of length \(h\) has variance \(h/4\) at its midpoint. Linear interpolation gives continuous functions \(B^{(k)}\). Since \(P(|Z|>c\sqrt k)\le e^{-c^2k/2}\) and there are \(2^k\) new points at stage \(k\), Borel–Cantelli shows that \(\max|B^{(k+1)}-B^{(k)}|\le c\sqrt{k}\,2^{-(k+2)/2}\) eventually, for \(c>\sqrt{2\ln2}\). The series converges uniformly, and the limit is a Brownian motion on \([0,1]\). The picture at the top of this page was drawn this way.

Brownian paths are thus Hölder continuous of every order \(\gamma<\tfrac12\) but not of order \(\tfrac12\). Compare Khinchin’s law of the iterated logarithm, which at a fixed time gives the smaller scale \(\sqrt{2h\log\log(1/h)}\). The difference measures how much worse the worst point is than a typical point.

Proof sketch (with Itô calculus). For fixed \(u\), Itô’s formula gives that \(Z_t=\exp\big(iuM_t+\tfrac12u^2t\big)\) is a bounded-on-compacts local martingale, hence a martingale. So \(\mathbb E[e^{iu(M_t-M_s)}\mid\mathcal F_s]=e^{-u^2(t-s)/2}\): the increment is \(N(0,t-s)\) and independent of \(\mathcal F_s\).

Arcsine laws, local time, the plane

For the general reader. In a long fair coin-tossing game, one player is usually ahead for most of the time, not half of it. Lévy found the exact law of this effect for Brownian motion. He also invented local time, a way to measure how much time a path spends at a single point, and studied Brownian motion in the plane, where he found deep links with complex analysis.

The idea goes back to Désiré André’s ballot problem and to Louis Bachelier; Lévy made it a standard tool. After the first hitting time of \(a\), the reflected path \(2a-B\) is again a Brownian motion, so paths that reach \(a\) end above or below \(a\) with equal probability.

The density is largest near 0 and 1: it is most likely that one side leads almost all the time. The law of the last zero is the easiest: \(g<x\) means no zero in \([x,1]\). Conditioning on \(B_x\) and using the reflection principle gives \(P(g<x)=\frac2\pi\arcsin\sqrt x\).

Local time

In the 1948 book Lévy introduced the local time at 0,

an increasing continuous process that grows only on the zero set of \(B\), a set of Lebesgue measure zero. In modern terms it appears in Tanaka’s formula \(|B_t|=\int_0^t\operatorname{sgn}(B_s)\,dB_s+L_t\).

Idea. Skorokhod’s lemma: for a continuous \(y\) with \(y(0)\ge 0\), there is a unique pair \((z,\ell)\) with \(z=y+\ell\ge0\), \(\ell\) increasing from 0 and growing only when \(z=0\); explicitly \(\ell_t=\max_{s\le t}(-y_s)^+\). Apply it to \(y=-B\), giving \((M-B,M)\), and to \(y=W\) from Exercise 9.1, giving \((|B|,L)\) by Tanaka’s formula. Since \(W\) and \(-B\) are both Brownian motions, the pairs have the same law.

Planar Brownian motion

In Le mouvement brownien plan (American Journal of Mathematics, 1940), Lévy showed that if \(Z\) is planar Brownian motion and \(f\) is analytic and non-constant, then \(f(Z)\) is a time-changed planar Brownian motion. He also studied the stochastic area \(\mathcal A_t=\tfrac12\int_0^t(X_s\,dY_s-Y_s\,dX_s)\) swept by the path, found its law, \(\mathbb E\,e^{iu\mathcal A_t}=1/\cosh(ut/2)\), and showed that planar Brownian paths have zero area as sets.

Polytechnique, war and hiding, 1920–1944

From 1920 Lévy taught analysis at the École Polytechnique and, until 1951, mathematics at the École des Mines. Neither school was a research institution, and he never held a Sorbonne chair or directed a large research group. His work in probability was done largely alone. French probabilists of the period, Émile Borel and Maurice Fréchet among them, were colleagues and correspondents; his correspondence with Fréchet, spanning five decades, is an important source on his thinking.

Lévy’s method was intuitive and geometric. He thought in terms of sample paths and wrote arguments that rigorous readers often found incomplete, which slowed acceptance of his work. When Kolmogorov’s axiomatic Grundbegriffe der Wahrscheinlichkeitsrechnung appeared in 1933, Lévy recognized that it gave probability the foundation he had not tried to build. Many of his results of the 1930s were found at about the same time by Kolmogorov, Khinchin, Feller and others, and priority questions occasionally arose.

His formal doctoral students were few. Wolfgang Doeblin, son of the novelist Alfred Döblin, wrote a thesis on Markov chains in 1938 under Fréchet and Lévy. Lévy admired him deeply. Doeblin died in June 1940 while serving in the French army, taking his own life to avoid capture, aged 25; a sealed envelope he had sent to the Académie des Sciences was opened in 2000 and showed that he had anticipated parts of Itô’s stochastic calculus. Michel Loève, who later built the Berkeley school of probability, completed his thesis with Lévy in 1941.

Vichy

In 1940 the Polytechnique moved to Lyon in the unoccupied zone. The Vichy government’s statute of 3 October 1940 excluded Jews from teaching, and Lévy was notified of his dismissal on 19 December 1940. After the director of the school intervened, he was allowed to teach again from 14 March 1941. In 1942, before the German occupation of the southern zone in November, he left Lyon and lived under a false identity near Grenoble, staying with his son-in-law Robert Piron at Montbonnot. He continued to work and to write to Fréchet until the Liberation, and then returned to his chair.

Late recognition and legacy, 1945–1971

After the war Lévy published his most original book, Processus stochastiques et mouvement brownien (1948; second edition 1965), with local time, the arcsine laws, planar Brownian motion and his characterization theorem. He suffered from tuberculosis, and in 1951 part of his left lung was removed. Benoit Mandelbrot completed his thesis in Paris in 1952 with Lévy among his mentors; Georges Matheron, the founder of geostatistics, finished in 1954. Lévy retired from the Polytechnique in 1959.

Recognition came late. He received the Émile Picard Medal of the Académie des Sciences in 1953, became an honorary member of the London Mathematical Society in 1963, and was elected to the Académie in 1964, at 77. In 1970 he published a memoir, Quelques aspects de la pensée d’un mathématicien. He died in Paris on 15 December 1971, aged 85.

Legacy

• Probability: Lévy processes, stable laws and infinitely divisible laws are a standard part of graduate probability. His martingale theorems prepared Doob’s systematic theory.
• Stochastic analysis: Kiyosi Itô named Lévy’s work on processes with independent increments as a starting point for his own; Lévy’s characterization and local time are basic tools of stochastic calculus.
• Applications: Mandelbrot used stable laws in 1963 to model cotton price changes, starting the study of heavy tails in finance. Lévy flights, random walks with stable jumps, are used in physics and in models of animal foraging.
• Geometry of high dimensions: Lévy’s isoperimetric inequality became, through Milman and Gromov, the concentration of measure phenomenon, central in Banach space theory, random matrices and theoretical computer science.
• Family: his daughter Marie-Hélène Schwartz worked on characteristic classes, and his son-in-law Laurent Schwartz won the Fields Medal in 1950 for the theory of distributions.

Connections

People linked to Paul Lévy across this collection. A name without a link has no page here yet. See Paul Lévy in the network of all 337 lives →

Teachers and advisors

• Jacques Hadamard 1911
• Vito Volterra

Students

• Benoit Mandelbrot 1952
• Georges Matheron 1954
• Michel Loève 1941
• Wolfgang Doeblin 1938

Influenced by

• Norbert Wiener
• René Gateaux

Influenced

• Joseph Doob
• Kiyosi Itô

Colleagues and collaborators

• Andrey Kolmogorov
• Henri Poincaré
• Maurice Fréchet
• Norbert Wiener
• Émile Borel
• Émile Picard
• Aleksandr Khinchin 1936

Family

• Laurent Schwartz
• Lucien Lévy
• Marie-Hélène Schwartz

Also mentioned in this life

• Stefan Banach
• Augustin-Louis Cauchy
• Pafnuty Chebyshev
• Joseph Fourier
• Mikhail Gromov
• Pierre-Simon Laplace
• Henri Lebesgue
• Aleksandr Lyapunov
• Andrey Markov
• Siméon Denis Poisson
• Bernhard Riemann

Fields

• probability theory 23 lives
• functional analysis 11 lives

On CoursesHub

• Functional Analysis · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Paul Lévy, MacTutor History of Mathematics.
2. Paul Lévy (mathematician), Wikipedia.
3. Paul Pierre Lévy, Mathematics Genealogy Project.
4. N. H. Bingham, Paul Lévy, 1886–1971, lecture slides, Imperial College.
5. M. Loève, Paul Lévy, 1886–1971, Annals of Probability 1 (1973), 1–18.
6. P. Lévy, Quelques aspects de la pensée d’un mathématicien, Blanchard, 1970.
7. B. Barbut, B. Locker, L. Mazliak, Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters, Springer, 2014.
8. Wolfgang Doeblin, Wikipedia.

Mathematics

1. P. Lévy, Sur les séries semi-convergentes, Nouvelles Annales de Mathématiques (1905).
2. P. Lévy, Leçons d’analyse fonctionnelle, Gauthier-Villars, 1922.
3. P. Lévy, Calcul des probabilités, Gauthier-Villars, 1925.
4. P. Lévy, Sur les intégrales dont les éléments sont des variables aléatoires indépendantes, Ann. Scuola Norm. Sup. Pisa (2) 3 (1934).
5. P. Lévy, Théorie de l’addition des variables aléatoires, Gauthier-Villars, 1937.
6. P. Lévy, Sur certains processus stochastiques homogènes, Compositio Mathematica 7 (1940).
7. P. Lévy, Le mouvement brownien plan, American Journal of Mathematics 62 (1940).
8. P. Lévy, Processus stochastiques et mouvement brownien, Gauthier-Villars, 1948.
9. J. Bertoin, Lévy Processes, Cambridge University Press, 1996.
10. K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999.
11. P. Mörters and Y. Peres, Brownian Motion, Cambridge University Press, 2010.
12. M. Ledoux, The Concentration of Measure Phenomenon, AMS, 2001.
13. D. Williams, Probability with Martingales, Cambridge University Press, 1991.