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\noindent{\bf Harald CRAM\'ER}\\
b. 25 September 1893 - d. 5 October 1985
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\noindent{\bf Summary.} Harald Cram\'er, mathematical master craftsman,
contributed pathbreaking
research in probability, statistics, and insurance mathematics, and to the
illumination of statistics as a coherent mathematical discipline.
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Harald Cram\'er was born in Stockholm, Sweden on September 25, 1893
and spent virtually his whole life there.
He entered the University of Stockholm in 1912, studying Chemistry and
Mathematics. He made several scientific contributions in Chemistry
with H.\ v.\ Euler before turning single mindedly to mathematics,
becoming a student of the influential Swedish mathematician Mittag-Leffler and
studying with the eminent Hungarian mathematician Marcel Riesz,
then visiting Mittag-Leffler's Institute. He wrote his doctoral
thesis on Dirichlet series, graduating in 1917, and published approximately
20 papers in analytic number theory over the next seven years.
This research stood him in good stead later, since in it he became familiar
with Fourier
integrals of a type closely related to those which were central to his
subsequent fundamental probabilistic contributions.
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\noindent
{\bf Insurance Mathematician.}
As reported in [4], a schism developed between
Cram\'er and the powerful but evidently difficult
Mittag-Leffler, who then strongly threatened
Cram\'er's career prospects in pure mathematics in Scandinavia.
As Cram\'er describes it, insurance companies provided a natural
alternative employment avenue for young Swedish mathematicians and
it was on such a career that he embarked in 1918 (also the year
of his marriage). The change of mathematical focus turned out
to be very fortunate in
Cram\'er's view, since it introduced him to both insurance mathematics and
associated probabilistic developments -- leading to his later wide ranging
probabilistic interests and contributions.
The insurance mathematics background of the time and
Cram\'er's contributions to the field are well summarized
in [4]. He quickly became one of the leading contributors to
the basic theory of insurance risks both through his own work and through
his influence on colleagues and students.
These contributions of course mathematically describe fundamental
mechanisms underlying insurance. But as noted in [4], he made
even more immediately practical contributions to insurance such as a treatment of
loadings in mortality assumptions. He provided a precise discussion of
the bias necessary in premium estimation and developed the
`zero point method' which was used for many years in setting insurance
rates.
Most of Cram\'er's works in insurance mathematics were published in the
1920-1935 period. However his interest in the field was lifelong
and he continued his contributions (both technical and educational)
well beyond that time even while doing pioneering work in other areas.
In particular his classic volume {\it Collective Risk Theory} was
published in 1955 and revised in 1968.
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\noindent
{\bf Probabilist.}
During the 1920's Cram\'er was led naturally to an interest
in probability through his work in insurance risk.
He refers to this period as a ``decade of preparation'' for the
explosive development of probability theory which was to follow,
but it nevertheless contained important contributions of his own,
as well as the blossoming developments by e.g.. Kolmogorov,
Wiener, L\'evy. Cram\'er had already begun using characteristic
functions in an extensive study of central limit theory,
and motivated by insurance applications he investigated the magnitude of
the error
in replacing a random sum by its normal limit. He gave an early
version of the bound later given final form by his student Esseen in his
1944 thesis, the famous ``Berry-Esseen Bound'', and detailed Edgeworth
(q.v.) expansions for the error.
In 1929 Cram\'er was appointed to a new chair in ``Actuarial
Mathematics and Mathematical
Statistics'' at Stockholm University, set up at the instigation of the
Swedish insurance companies. The institute became very
active under his leadership and many of the now celebrated results
of Cram\'er himself, and his students and colleagues
emanated from it in the ensuing 20 years. The decade of the 30's
(described by Cram\'er as the ``Heroic Period'') saw
dramatic developments in central limit and related theory, involving
the Stockholm group (including Feller at that time) as
well as that of Khintchine, Wiener, L\'evy and others.
Cram\'er became deeply involved in the developing early theory of
stationary processes. In particular he generalized the (1934) covariance
representations of Khintchine to a vector context, and the 1938 thesis
of his then student H.
Wold contained the renowned decomposition of a stationary
sequence into its deterministic and purely non-deterministic
components.
The years of the second world war caused professional isolation
but Cram\'er maintained a remarkable level of activity at the
Stockholm Institute. For example the (1940) thesis by O. Lundberg
(q.v.) on stochastic
models provided an actuarial basis for non-life insurance
which was widely used. He made the most of the few available
opportunities for
international contacts. One of these was a primarily Scandinavian
conference he
organized in Stockholm in 1941 at which he presented the spectral
representation of a stationary process (surely one
of his most remarkable and exciting results -- also obtained independently
by M. Lo\`eve in the war years).
In the postwar 1940's Cram\'er's Institute continued its strong stochastic
process activity. This included fundamental work of
Karhunen on stationary processes, the pathbreaking thesis of
Cram\'er's then student Grenander involving inference for stochastic
processes and the work of
Cram\'er himself on general spectral representations for classes
of nonstationary processes.
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\noindent
{\bf Statistician.}
``Probability'' is surely the life blood of
``statistics'' and any dividing line is
impossible to position. This is especially true of
Cram\'er's work, much of which may be regarded as
``Mathematical Methods of Statistics'', as indicated by the
title to his most celebrated volume. This enormously influential work
was written as a way of making good use of the period of enforced
isolation caused by the second world war was described by
Cram\'er as his ``Calling card to the postwar scientific world''
([4]). It provided a wonderfully timely and lucid account of a
hitherto hodge podge of often mysterious statistical procedures,
now organized as a coherent mathematical discipline.
It does of course contain many of its author's own contributions,
including publication of the celebrated
`Cram\'er--Rao' bound discovered independently by Rao (1945)
{\it Mathematical Methods} has remarkably few blemishes
(one being the constructive definition of Borel sets) and may be perhaps aptly
described as ``almost uniformly error free''. It has had
immense influence on generations of statisticians and especially, I would
conjecture, in encouraging young (myself included at
the time!) mathematicians to enter and find a mathematically
satisfying career in
statistics and its applications.
One could not describe {\it Mathematical Methods} as a manual of
applied statistics. But it clearly highlights the fact that
Cram\'er was substantially motivated in his mathematical work by
the potential for applications. Indeed, as noted in [1], he
actually attempted to set up an applied statistics section in the Stockholm
group, a plan which did not reach full fruition though a less formal applied
activity was instituted.
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\noindent
{\bf Administrative, and later years.}
In 1950
Cram\'er was elected President of Stockholm University and in
1958 Chancellor of the entire Swedish university system. This
administrative service undoubtedly had a severe impact on his
scientific production but by no means halted it, as can be seen from his
publications of the period. On his retirement from the Chancellorship
in 1961 he again became very active in research.
He made several extended visits to the U.S.\ to participate in
research activities at the Research Triangle Institute (RTI)
at the invitation of Gertrude Cox (q.v.), and to give lectures.
I first met Harald Cram\'er at RTI as we worked together there on a project
involving the reliability of spacecraft guidance systems, under a
contract with NASA. This sparked his interest in
level-crossing and extremal problems for stationary processes in which
he made important contributions -- on which we wrote a joint book [3].
This again underscored his love for development of
mathematics relevant to applications.
Simultaneously
Cram\'er investigated questions of ``multiplicity theory'' and innovation
representations'' for vector stochastic processes. He wrote a series
of approximately 10 basic papers in this area from 1960, the last
being published in 1982, at age 89!
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\noindent
{\bf Some personal comments.}
I have focussed in this biography on Cram\'er the scientist, and his
pioneering research contributions, but there is much to be learned from
other sides to his life and work, dealt with e.g. in
[1, 4, 5].
Harald
Cram\'er had few peers as teacher and educator, through his written works
and incomparably clear classroom teaching, formal lectures and
personal interactions. The direct application of his research work is perhaps
clearest in the insurance context. But, as indicated throughout, his
fundamental theoretical work substantially arose from the need to provide
methods for dealing with real applied problems. Thus he strongly facilitated
the use of probability and statistics and their wide ranging impact on society.
His enormous personal educational impact is surely now less direct but
nevertheless has permanently influenced the field.
Cram\'er's gentlemanly demeanor, courtesy and concern
for others were a counter-example to the thesis that professional
greatness must be accompanied by some form of social or character
weakness. Finally, no record of the life of Harald
Cram\'er should omit mention of his wife Marta -- his lifelong
companion from their 1918 marriage until her death in 1973.
No doubt Cram\'er would have made pioneering contributions if he had remained single
or had a less successful marriage. But the extraordinarily
fine mutual support and life partnership with his
``Beloved Marta'', and their strong family relationships
gave an extra quality to his -- indeed their -- professional and
human legacy.
Harald Cram\'er died in his native Stockholm on
October 5, 1985, with the conviction he expressed just prior to his death, that
his life's work was now complete.
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\noindent
{\bf Further reading}
Several detailed accounts are available of the life and
work of Harald Cram\'er
and his fundamental contributions in [1, 4, 6, 5, 7]. Especially
illuminating is
Cram\'er's own description in [2]
which gives a fascinating account of this exciting period in
probability and statistical science and his role in it.
A reader interested in the details of
Cram\'er's contributions and their motivating environment
will be rewarded by study of [2] and the other references cited. In
particular, [1] contains a comprehensive bibliography of his published works.
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\begin{thebibliography}{3}
\bibitem{1} [1] Blom, G. (1987). Harald Cram\'er 1893-1985. {\em Annals
of Statistics}, {\bf 15},
1335-1350.
\bibitem{2} [2] Cram\'er, H.\ (1976). Half a century with probability theory: Some personal
recollections. {\em Annals of Probability}, {\bf 4}, 509-546.
\bibitem{3} [3] Cram\'er, H.\ \& Leadbetter, M.R.\ (1967). {\em Stationary and Related
Stochastic Processes}. Wiley, New York.
\bibitem{4} [4] Grenander, U.\ (1994). A survey of the life and works of Harald Cram\'er,
{\em Scandinavian Actuarial Journal}, (1995) {\bf 1}, 2-5.
\bibitem{5} [5] Leadbetter, M.R.\ (1988). Harald Cram\'er 1893-1985. {\em
International Statistical
Review}, {\bf 56}, 89-97.
\bibitem{6} [6] Martin-L\"of, A.\ (ed.) (1994). {\em Harald Cram\'er: Collected works}.
Vols.\ I \& II, Springer-Verlag, Berlin.
\bibitem{7} [7] Leadbetter, M.R. (1995). A view of Harald Cram\'er: working with the master in his
later years. {\em Scandinavian Actuarial Journal}, {\bf 1}, 6-11.
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\hfill{M.R. Leadbetter}
\end{thebibliography}
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