Harald Cramér
Encyclopedia
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".
, Sweden
on September 25, 1893. Cramér remained close to Stockholm for most of his life. He entered the University of Stockholm as an undergraduate in 1912, where he studied mathematics
and chemistry
. During this period, he was a research assistant under the famous chemist, Hans von Euler-Chelpin
, with whom he published his first 5 articles from 1913–1914. Following his lab experience, he began to focus solely on mathematics
. He eventually began his work on his doctoral studies in mathematics which were supervised by the famous Hungarian mathematician, Marcel Riesz
, at the University of Stockholm. Also influenced by G.H. Hardy, Cramér's research led to a PhD in 1917 for his thesis "On a class of Dirichlet series".
. He also made some important statistical contributions to the distribution of primes and twin primes. His most famous paper on this subject is entitled "On the order of magnitude of the difference between prime numbers", which provided a rigorous account of the constructive role in which probability
applied to number theory
.
In the late 1920s, Cramér became interested in the field of probability, which at the time was not an accepted branch of mathematics. Cramér knew that a radical change was needed in this field, and in a paper in 1926 said, "The probability concept should be introduced by a purely mathematical definition, from which its fundamental properties and the classical theorems are deduced by purely mathematical operations." Cramér took an interest in the rigorous mathematical formulation of probability in the work of French
and Russian
mathematicians such as Kolmogorov, Lévy, Bernstein, and Khinchin in the early 1930s. Cramér also made significant development to the revolution in probability theory.
Cramér later wrote his careful study of the field in his Cambridge publication Random variables and probability distributions which appeared in 1937. Shortly after World War II, Cramér would go on to publish the influential Mathematical Methods of Statistics in 1946. This text was one that "showed the way in which statistical practice depended on a body of rigorous mathematical analysis as well as Fisherian intuition."
In 1929, Cramér was appointed to a newly created chair in Stockholm University, becoming the first Swedish professor of Actuarial Mathematics and Mathematical Statistics. Cramér retained this position up until 1958. During his tenure at Stockholm University, Cramér was a PhD advisor for 10 students, most notably Herman Wold and Kai Lai Chung. Starting in 1950, Cramér took on the additional responsibility of becoming the President of Stockholm University. In 1958, he was also appointed to be Chancellor of the entire Swedish university system. Dr. Cramér would eventually retire from the Swedish university system in 1961.
and insurance mathematics. During the period from 1920 to 1929, he was an actuary for the life insurance company Svenska livförsäkringsbolaget. His actuarial work during this time led him to study probability and statistics which became the main area of his research. In 1927 he published an elementary text in Swedish Probability theory and some of its applications. Following his work for Svenska livförsäkringsbolaget, he went on to work for Återförsäkringsaktiebolaget Sverige, a reinsurance
company, up until 1948. He was also known for his pioneering efforts in insurance risk theory
. After this period, he remained as a consultant actuary to Sverige from 1949 to 1961. Later in his life, he was elected to be the Honorary President of the Swedish Actuarial Society.
Cramér remained an active contributor to his professional career for an additional 20 years. Following his retirement in 1961, he became extremely active in research, which had been slowed due to his Chancellorship. During the years from 1961 to 1983, Cramér traveled throughout the United States and Europe to continue his research, making significant stops at Berkeley, Princeton, and at the Research Triangle Institute of North Carolina. His academic career spanned over seven decades, from 1913 to 1982.
Harald Cramér married Marta Hansson in 1918, and they remained together up until her death in 1973. He had often referred to her as his "Beloved Marta". Together they had one daughter, Marie-Louise, and two sons, Tomas and Kim.
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....
, actuary
Actuary
An actuary is a business professional who deals with the financial impact of risk and uncertainty. Actuaries provide expert assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms ....
, and statistician
Statistician
A statistician is someone who works with theoretical or applied statistics. The profession exists in both the private and public sectors. The core of that work is to measure, interpret, and describe the world and human activity patterns within it...
, specializing in 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 probabilistic number theory
Probabilistic number theory
Probabilistic number theory is a subfield of number theory, which explicitly uses probability to answer questions of number theory. One basic idea underlying it is that different prime numbers are, in some serious sense, like independent random variables...
. He was once described by John Kingman
John Kingman
Sir John Frank Charles Kingman, born on 28 August 1939 in Beckenham, Kent, is a British mathematician.He was N. M. Rothschild and Sons Professor of Mathematical Sciences and Director of the Isaac Newton Institute at the University of Cambridge from 2001 until 2006, when he was succeeded by Sir...
as "one of the giants of statistical theory".
Early life
Harald Cramér was born in StockholmStockholm
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...
, Sweden
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....
on September 25, 1893. Cramér remained close to Stockholm for most of his life. He entered the University of Stockholm as an undergraduate in 1912, where 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...
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....
. During this period, he was a research assistant under the famous chemist, Hans von Euler-Chelpin
Hans von Euler-Chelpin
Hans Karl August Simon von Euler-Chelpin was a German-born Swedish biochemist. He won the Nobel Prize in Chemistry in 1929 with Arthur Harden for their investigations on the fermentation of sugar and fermentative enzymes.He was professor of general and organic chemistry at Stockholm University...
, with whom he published his first 5 articles from 1913–1914. Following his lab experience, he began to focus solely on 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...
. He eventually began his work on his doctoral studies in mathematics which were supervised by the famous Hungarian mathematician, Marcel Riesz
Marcel Riesz
Marcel Riesz was a Hungarian mathematician who was born in Győr, Hungary . He moved to Sweden in 1908 and spent the rest of his life there, dying in Lund, where he was a professor from 1926 at Lund University...
, at the University of Stockholm. Also influenced by G.H. Hardy, Cramér's research led to a PhD in 1917 for his thesis "On a class of Dirichlet series".
Academic professional career
Following his PhD, he served as an Assistant Professor of Mathematics at Stockholm University from 1917 to 1929. Early on, Cramér was highly involved in analytic number theoryAnalytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
. He also made some important statistical contributions to the distribution of primes and twin primes. His most famous paper on this subject is entitled "On the order of magnitude of the difference between prime numbers", which provided a rigorous account of the constructive role in which 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...
applied to 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...
.
In the late 1920s, Cramér became interested in the field of probability, which at the time was not an accepted branch of mathematics. Cramér knew that a radical change was needed in this field, and in a paper in 1926 said, "The probability concept should be introduced by a purely mathematical definition, from which its fundamental properties and the classical theorems are deduced by purely mathematical operations." Cramér took an interest in the rigorous mathematical formulation of probability in the work of French
French people
The French are a nation that share a common French culture and speak the French language as a mother tongue. Historically, the French population are descended from peoples of Celtic, Latin and Germanic origin, and are today a mixture of several ethnic groups...
and Russian
Russians
The Russian people are an East Slavic ethnic group native to Russia, speaking the Russian language and primarily living in Russia and neighboring countries....
mathematicians such as Kolmogorov, Lévy, Bernstein, and Khinchin in the early 1930s. Cramér also made significant development to the revolution in probability theory.
Cramér later wrote his careful study of the field in his Cambridge publication Random variables and probability distributions which appeared in 1937. Shortly after World War II, Cramér would go on to publish the influential Mathematical Methods of Statistics in 1946. This text was one that "showed the way in which statistical practice depended on a body of rigorous mathematical analysis as well as Fisherian intuition."
In 1929, Cramér was appointed to a newly created chair in Stockholm University, becoming the first Swedish professor of Actuarial Mathematics and Mathematical Statistics. Cramér retained this position up until 1958. During his tenure at Stockholm University, Cramér was a PhD advisor for 10 students, most notably Herman Wold and Kai Lai Chung. Starting in 1950, Cramér took on the additional responsibility of becoming the President of Stockholm University. In 1958, he was also appointed to be Chancellor of the entire Swedish university system. Dr. Cramér would eventually retire from the Swedish university system in 1961.
Actuarial career
A large portion of Cramér's work concerned the field of actuarial scienceActuarial science
Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries. Actuaries are professionals who are qualified in this field through education and experience...
and insurance mathematics. During the period from 1920 to 1929, he was an actuary for the life insurance company Svenska livförsäkringsbolaget. His actuarial work during this time led him to study probability and statistics which became the main area of his research. In 1927 he published an elementary text in Swedish Probability theory and some of its applications. Following his work for Svenska livförsäkringsbolaget, he went on to work for Återförsäkringsaktiebolaget Sverige, a reinsurance
Reinsurance
Reinsurance is insurance that is purchased by an insurance company from another insurance company as a means of risk management...
company, up until 1948. He was also known for his pioneering efforts in insurance risk theory
Risk theory
Risk theory connotes the study usually by actuaries and insurers of the financial impact on a carrier of a portfolio of insurance policies. For example, if the carrier has 100 policies that insures against a total loss of $1000, and if each policy's chance of loss is independent and has a...
. After this period, he remained as a consultant actuary to Sverige from 1949 to 1961. Later in his life, he was elected to be the Honorary President of the Swedish Actuarial Society.
Personality and later years
Harald Cramér married Marta Hansson in 1918, and they remained together up until her death in 1973. He had often referred to her as his "Beloved Marta". Together they had one daughter, Marie-Louise, and two sons, Tomas and Kim.
See also
- Cramér's conjecture
- Cramér's theoremCramér's theoremIn mathematical statistics, Cramér's theorem is one of several theorems of Harald Cramér, a Swedish statistician and probabilist.- Normal random variables :...
- Cramér–Rao bound
- Cramér's V
- Cramér–Wold theorem
- Large deviations theoryLarge deviations theoryIn probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. Some basic ideas of the theory can be tracked back to Laplace and Cramér, although a clear unified formal definition was introduced in 1966 by Varadhan...
- Cramérs inequality
External links
- photographs of Harald Cramér at the Oberwolfach Photo Collection
- http://www.york.ac.uk/depts/maths/histstat/people/cramer.gifPhotograph with C. R. RaoC. R. RaoCalyampudi Radhakrishna Rao FRS known as C R Rao is an Indian statistician. He is currently professor emeritus at Penn State University and Research Professor at the University at Buffalo. Rao has been honored by numerous colloquia, honorary degrees, and festschrifts and was awarded the US...
] from the PORTRAITS OF STATISTICIANS
