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\noindent{\bf Mikhailo Pylypovych KRAVCHUK (or KRAWTCHOUK)}\\
b. 27 September 1892 (o.s.) - d. 9 March 1942
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{\bf Summary.} Kravchuk was a Professor of Mathematical Statistics in Kyiv,
Ukraine. He discovered the ``Krawtchouk" polynomials, and made penetrating
contributions to various branches of mathematics before his arrest and death.
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Kravchuk (``Krawtchouk" is a transliteration he used when writing in French) was born in
the village
of Chovnytsia, in the Volyn (Volhynia) region of Ukraine.
After secondary studies at the gymnasium at Lutsk, Kravchuk entered the
Physico-Mathematical Faculty at Kyiv (Kiev) University, where in 1916 the algebraist D.O.
Grav\`{e}
described him as one of his most capable students. After a turbulent period on account of
World War
1, the revolution and civil war, Kravchuk worked as professor in charge of mathematics and
variability statistics in the Kyiv Agricultural Institute (1921-1929), in whose {\it Zapysky}
({\it
Memoirs}) the Krawtchouk polynomials, a system of polynomials orthonormal with respect to the
binomial distribution, make their first appearance in 1929.
He worked simultaneously at a number of other institutions: his first
statistical publication appears in 1925 in the {\it Zapysky} of the
Veterinary-Zootechnical Institute; and at the 1928 Bologna International
Congress of Mathematicians he is listed as representing Ukraine from the
National Institute of Education. In this connection especially he fostered
the use of the Ukrainian language in Ukrainian mathematics, was involved in
writing school texts, and in the compilation of a Ukrainian mathematical
dictionary. The majority of his publications in mathematical statistics,
listed and described by Movchan (1972), are in Ukrainian. An abbreviated
version of the contents of the 1929 Ukrainian-language paper, without proofs,
together with a summary of some additional material to appear in 1931 appeared
in French ({\it C.R. Acad. Sci., Paris}, {\bf 189} (1929) 620-622). It is
through this version and the definitive treatment of it in Szeg\"{o} (1939) that
Kravchuk's name in its French form was attached to the orthonormal polynomial
system. This system, with the aftermath of its creation including the work of
Kravchuk's students and associates (O.S. Smohorshewsky, S.M. Kulik, O.K.
Lebedintseva) and overseas work, are excellently treated in English in Prizva
(1992). This survey also contains a quite detailed description of the contents
of a major paper in mathematical statistics of 1931 published after Kravchuk's
election to the Ukrainian Academy on 29 June 1929, which has Ukrainian title
``Orthogonal polynomials associated with sampling with and without replacement"
and French title ``Sur les polyn\^{o}mes orthogonaux li\'{e}s avec les
sch\'{e}mas de Bernoulli et de C. Pearson", there being a long French summary
of some 9 pages. We return to it below.
Kravchuk was a mathematician of enormous breadth (mathematical statistics was
only one of the areas in which he made significant contributions) and energy
(some 180 publications in the period 1914-1938 in addition to heavy commitments
in the various scientific insitutions). In addition to his work in algebra
under the influence of Grav\`{e}, he passed on to functions of a real variable
and other areas of analysis, differential and integral equations, and geometry
as well as mathematical statistics and probability. In the period 1934-1938 he
headed the Department of Mathematical Statistics of the Institute of
Mathematics of the Ukrainian Academy of Science. His work in mathematical
statistics relates to the theory of correlation and regression, the bivariate
normal, the method of moments in statistics, and orthogonal polynomial systems.
Some of this was influenced by the work of the English Biometric School headed
by Karl Pearson (q.v.), as was the early work of Slutsky (q.v.), also in Kyiv in the
early years of his career.
A theme running through much of Kravchuk's work, and not only in mathematical
statistics, is the method of moments. His statistical direction was initially
driven by the work of Chebyshev (q.v.) on interpolation and the method of
moments, which in turn derived partly from Chebyshev's contact with the works
of Bienaym\'{e} (q.v.). In the {\it Zapysky} version of the 1929 paper, all
the substantial papers of Chebyshev on interpolation are mentioned. Kravchuk,
however, uses a more modern matrix approach in contrast to Chebyshev's continued
fractions. Chebyshev's work was dominant in probability and statistics in the
old czarist Russian empire, and this influence of the St. Petersburg School is
natural on a strong mathematician such as Kravchuk. The increasing interest in
statistics itself surely derived from Kravchuk's work in institutions such as those
whose {\it Zapysky} we have mentioned, where the need for application of
statistical methods was transparent.
Kravchuk viewed his polynomial system as a generalization of both the
Hermite polynomials (orthonormal with respect to the standard normal) and the
Charlier polynomials (orthonormal with respect to the Poisson), inasmuch as
both could be obtained in the limit from his. His motivation for studying them
was the fact that Chebyshev's polynomials (viewed as orthonormal with respect to
the uniform distribution on $0, 1, \cdots, n-1$) were a discrete analogue of the
Legendre polynomials, which are obtainable in the limit. A little-noticed feature
of Kravchuk's work is the expansion of a binomial probability value in terms of
Krawtchouk polynomials orthonormal with respect to another binomial
distribution. The germ of the idea is the analogous expansion of densities in
terms of Hermite polynomials in Chebyshev's last probability paper (1887); and
Kravchuk's expansion also produces the Charlier expansion of a binomial probability
in the limit. More importantly, he gives a bound on the error of partial expansion,
a very Chebyshevian idea. More importantly still, in his 1931 paper, he produces a
polynomial system orthonormal with respect to the hypergeometric distribution.
Subsequently this system, overlooked at the time outside Ukraine, even though it was
described by Smohorshewsky in the very same French journal ({\it C.R. Acad. Sci., Paris},
{\bf 200} (1935) 801-803)
has come to be
credited to later work of W. Hahn. Inasmuch as the binomial distribution can be
considered as a limit of the hypergeometric, this system generalizes the Krawtchouk
polynomials.
Kravchuk was arrested on the 21 February 1938 on a number of false charges,
primarily Ukrainian bourgeois nationalism and encouraging foreign espionage
because of his contacts with mathematicians outside the Soviet Union,
particularly with Ukrainians in Western Ukraine (specifically Levitsky and
Chaikovsky, in L'viv), then a part of Poland. He died near Magadan in Siberia.
Plaques in his memory have now been erected in Lutsk and in Kyiv, and a small
museum established in part of the elementary school in Chovnytsia.
An important and well-organized conference dedicated to his memory
was held in Kyiv and Lutsk on the 50th anniversary of his death and there was a
special number of the {\it Ukrainian Mathematical Journal} {\bf 44}, No.7,
1992. There is some biographical material about Kravchuk in Ukrainian, but little
in English or Russian. For a number of years after his arrest he was officially a
non-person in the Soviet Union. A general summary of his work, in English, is
given by Parasyuk and Virchenko (1992). A brief sketch in English of his life and
probabilistic work is given in Seneta (1993).
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\begin{thebibliography}{3}
\bibitem{1}
Movchan, V.O. (1972). Mathematical statistics and probability theory in the works
of M.P. Kravchuk. (In Ukrainian.) {\em Narysy Istor. Pryrodoznav.
i Tekh.,} {\bf 17}, 8-15.
\bibitem{2}
Parasyuk, O.S. and Virchenko, N.O. (1992). A short piece about the scientific heritage
of M. Kravchuk. {\em Ukrainian Mathematical
Journal,} {\bf 44}, 772-791.
\bibitem{3}
Prizva, G.I. (1992). Orthogonal Kravchuk polynomials.{\em Ukrainian Mathematical
Journal}, {\bf 44}, 792-800.
\bibitem{4}
Seneta, E.(1993). Krawtchouk polynomials and Australian statisticians. {\em Institute
of Mathematical Statistics. Bulletin.} {\bf 22}, No.4, July/August,
421-423.
\bibitem{5}
Szeg\"{o}, G. (1939). {\em Orthogonal Polynomials}. American Mathematical Society
Colloquium Publications, Vol.23.
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\hfill E. Seneta
\end{thebibliography}
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