Carl-Gustav Esseen
Encyclopedia
Carl-Gustav Esseen was a Swedish
mathematician
. His work was in the theory of probability
. The Berry–Esseen theorem
is named after him.
. Starting in 1936, he studied mathematics
, astronomy
, physics
and chemistry
at the University of Uppsala. Inspired by the harmonic-analytic
research of Harald Cramér
and Arne Beurling
, Esseen examined the accuracy of the approximation
to the normal distribution in the central limit theorem
in the case of independent and identically distributed summand
s. Esseen's bound is now called "the Berry-Esseen theorem", because it was independently proved by Andrew C. Berry, also.
In 1944 Esseen received his doctorate with a thesis on the Fourier analysis of probability distribution
s. In 1949 he was appointed full professor of applied mathematics
at the Royal Institute of Technology
in Stockholm
. In 1962 his professorship moved to the field of mathematical statistics
and in 1967, he became first holder of the chair of mathematical statistics at the University of Uppsala. He retired in 1984.
and related topics, he also worked in other areas. Some industrial applications were considered in his writings, for example, his studies on control theory and in telecommunications. After retirement, Esseen worked on topics from number theory
, especially factorization
, a topic of importance in cryptology.
Esseen supervised several doctoral students. His lectures and writings were meticulously prepared and delivered.
Sweden
Sweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. His work was in the theory of probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
. The Berry–Esseen theorem
Berry–Esséen theorem
The central limit theorem in probability theory and statistics states that under certain circumstances the sample mean, considered as a random quantity, becomes more normally distributed as the sample size is increased...
is named after him.
Life
Carl-Gustav Esseen attended school in LinköpingLinköping
Linköping is a city in southern middle Sweden, with 104 232 inhabitants in 2010. It is the seat of Linköping Municipality with 146 736 inhabitants and the capital of Östergötland County...
. Starting in 1936, he studied mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
at the University of Uppsala. Inspired by the harmonic-analytic
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
research of Harald Cramér
Harald Cramér
Harald Cramér was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. He was once described by John Kingman as "one of the giants of statistical theory".-Early life:Harald Cramér was born in Stockholm, Sweden on September...
and Arne Beurling
Arne Beurling
Arne Carl-August Beurling was a Swedish mathematician and professor of mathematics at Uppsala University and later at the Institute for Advanced Study in Princeton, New Jersey....
, Esseen examined the accuracy of the approximation
Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...
to the normal distribution in the central limit theorem
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
in the case of independent and identically distributed summand
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s. Esseen's bound is now called "the Berry-Esseen theorem", because it was independently proved by Andrew C. Berry, also.
In 1944 Esseen received his doctorate with a thesis on the Fourier analysis of probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s. In 1949 he was appointed full professor of applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
at the Royal Institute of Technology
Royal Institute of Technology
The Royal Institute of Technology is a university in Stockholm, Sweden. KTH was founded in 1827 as Sweden's first polytechnic and is one of Scandinavia's largest institutions of higher education in technology. KTH accounts for one-third of Sweden’s technical research and engineering education...
in Stockholm
Stockholm
Stockholm is the capital and the largest city of Sweden and constitutes the most populated urban area in Scandinavia. Stockholm is the most populous city in Sweden, with a population of 851,155 in the municipality , 1.37 million in the urban area , and around 2.1 million in the metropolitan area...
. In 1962 his professorship moved to the field of mathematical statistics
Mathematical statistics
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis...
and in 1967, he became first holder of the chair of mathematical statistics at the University of Uppsala. He retired in 1984.
Scientific work
Although Esseen worked mostly on the central limit theoremCentral limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
and related topics, he also worked in other areas. Some industrial applications were considered in his writings, for example, his studies on control theory and in telecommunications. After retirement, Esseen worked on topics from number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, especially factorization
Integer factorization
In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
, a topic of importance in cryptology.
Esseen supervised several doctoral students. His lectures and writings were meticulously prepared and delivered.
Selected works
- "On the Liapounoff limit of error in the theory of probability", Arkiv för Mathematik, Astronomi och Fysik 28A, #9 (1942), 19 pp., .
- "Determination of the maximum deviation from the Gaussian law", Arkiv för Mathematik, Astronomi och Fysik 29A, #20 (1943), 10 pp., .
- "Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law", Acta MathematicaActa MathematicaActa Mathematica is a journal publishing original research papers in all fields of mathematics. The journal was founded by Gösta Mittag-Leffler in 1882 and is published by Institut Mittag-Leffler, a research institute for mathematics belonging to the Royal Swedish Academy of Sciences...
77, #1 (December, 1945), pp. 1–125, , . (Dissertation) - "On mean central limit theorems", Kungl. Tekn. Högsk. Handl. Stockholm #121 (1958), 31 pp., .
- "Bounds for the absolute third moment", Scandinavian Journal of Statistics 2, #3 (1975), pp. 149–152, , .
- (with Svante JansonSvante JansonSvante Janson is a Swedish mathematician. A member of the Royal Swedish Academy of Sciences since 1994, Janson has been the chaired professor of mathematics at Uppsala University since 1987....
) "On moment conditions for normed sums of independent variables and martingale differences", Stochastic Processes and their Applications 19, #1 (1985), pp. 173–182, , . - "A stochastic model for primitive roots", Revue Roumaine de Mathématiques Pures et Appliquées 38, #6 (1993), pp. 481–501, .