Harald Cramér
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Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
,
actuary An actuary is a professional with advanced mathematical skills who deals with the measurement and management of risk and uncertainty. These risks can affect both sides of the balance sheet and require investment management, asset management, ...
, and
statistician A statistician is a person who works with Theory, theoretical or applied statistics. The profession exists in both the private sector, private and public sectors. It is common to combine statistical knowledge with expertise in other subjects, a ...
, specializing in
mathematical statistics Mathematical statistics is the application of probability theory and other mathematical concepts to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques that are commonly used in statistics inc ...
and probabilistic number theory. John Kingman described him as "one of the giants of statistical theory".Kingman 1986, p. 186.


Biography


Early life

Harald Cramér was born in
Stockholm Stockholm (; ) is the Capital city, capital and List of urban areas in Sweden by population, most populous city of Sweden, as well as the List of urban areas in the Nordic countries, largest urban area in the Nordic countries. Approximately ...
,
Sweden Sweden, formally the Kingdom of Sweden, is a Nordic countries, Nordic country located on the Scandinavian Peninsula in Northern Europe. It borders Norway to the west and north, and Finland to the east. At , Sweden is the largest Nordic count ...
on 25 September 1893. Cramér remained close to Stockholm for most of his life. He entered the
Stockholm University Stockholm University (SU) () is a public university, public research university in Stockholm, Sweden, founded as a college in 1878, with university status since 1960. With over 33,000 students at four different faculties: law, humanities, social ...
as an undergraduate in 1912, where he studied
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
. During this period, he was a research assistant under the famous chemist, Hans von Euler-Chelpin, with whom he published his first five articles from 1913 to 1914. Following his lab experience, he began to focus solely on
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. He eventually began his work on his doctoral studies in mathematics which were supervised by Marcel Riesz at the Stockholm University. Also influenced by
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
, Cramér's research led to a PhD in 1917 for his thesis "On a class of
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in anal ...
".


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 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 Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...
. He also made some important statistical contributions to the distribution of primes and
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime' ...
s. His most famous paper on this subject is entitled "On the order of magnitude of the difference between consecutive prime numbers", which provided a rigorous account of the constructive role in which
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
applied to
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
Kendall 1986, p. 211. and included an estimate for prime gaps that became known as Cramér's conjecture. 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."Blom 1987, p. 1340. 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 (with a 2nd edition in 1962 and a 3rd edition in 1970). Shortly after World War II, Cramér went 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." His 1955 book ''Elements of Probability Theory and Some of its Applications'' introduces probability theory at a more elementary level than ''Mathematical Methods of Statistics''. 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. In 1950 he was elected as a Fellow of the American Statistical Association. 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. Cramér retired from the Swedish university system in 1961.


Actuarial career

A large portion of Cramér's work concerned the field of
actuarial science Actuarial science is the discipline that applies mathematics, mathematical and statistics, statistical methods to Risk assessment, assess risk in insurance, pension, finance, investment and other industries and professions. Actuary, Actuaries a ...
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 is insurance that an insurance company purchases from another insurance company to insulate itself (at least in part) from the risk of a major claims event. With reinsurance, the company passes on ("cedes") some part of its own insu ...
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.


Later years

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. Cramér received an
Honorary Doctorate An honorary degree is an academic degree for which a university (or other degree-awarding institution) has waived all of the usual requirements. It is also known by the Latin phrases ''honoris causa'' ("for the sake of the honour") or '' ad hon ...
from
Heriot-Watt University Heriot-Watt University () is a public research university based in Edinburgh, Scotland. It was established in 1821 as the School of Arts of Edinburgh, the world's first mechanics' institute, and was subsequently granted university status by roya ...
in 1972. His academic career spanned over seven decades, from 1913 to 1982.Blom 1987, p. 1338. 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.


Notes


References

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External links

*
photographs of Harald Cramér
at th
Oberwolfach Photo CollectionPhotograph
with C. R. Rao from th
PORTRAITS OF STATISTICIANS
* {{DEFAULTSORT:Cramer, Harald 1893 births 1985 deaths Swedish statisticians Swedish actuaries Probability theorists 20th-century Swedish mathematicians Scientists from Stockholm Fellows of the American Statistical Association Foreign associates of the National Academy of Sciences Members of the Royal Society of Sciences in Uppsala Mathematical statisticians