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\noindent{\bf Andrei Andreevich MARKOV}\\
b. 2 June 1856 (o.s.) - d. 20 July 1922
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\noindent
{\bf Summary.} Markov, with Liapunov a disciple of Chebyshev, gave rigorous
proofs of the Central Limit Theorem. Through his work on Markov chains,
the concept of Markovian dependence pervades modern theory and application
of random processes. His textbook influenced the development of probability
and statistics internationally.
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Markov was born in Ryazan, and died in Petrograd (which was - before the
revolution, and is now again - called St. Petersburg). He was a poor student
in all but mathematics at the fifth Petersburg gymnasium which he entered in
1866. Already during this period he revealed an emotional and uncompromising
nature which was to surface in clashes with the tsarist regime and academic
colleagues, even though his motives were generally high-minded. He was,
however, more fortunate in his circumstances than his similarly volatile
younger countryman E.E. Slutsky (q.v.) in that Markov had influential senior
colleagues who understood and tolerated him, among whom V.A. Steklov is
mentioned frequently; and in that he worked in the capital city of the Russian
Empire.
Entering Petersburg University in 1874, he attended classes in the
Physico-Mathematical Faculty by A.N. Korkin, E.I. Zolotarev and P.L. Chebyshev
(q.v.), all of whom encouraged him and facilitated his progress. At the
completion of his studies in 1878 he received a gold medal and was retained by
the university to prepare for a career as an academic, in the tradition of the
times with the best students.
With the departure of Chebyshev from the university in 1883, Markov took over
his course in probability which he continued to teach yearly. Markov's
doctoral dissertation {\it On Some Applications of Algebraic Continued
Fractions}, results from which were published in 1884, already had implicit
connections with probability theory inasmuch as it treated certain inequalities
published by Chebyshev in 1874 in Liouville's {\it J. Math. Pures Appl.},
relating to the method of moments which Chebyshev had in turn extracted from
notions of I.J. Bienaym\'{e} (q.v.). At the proposal of Chebyshev, Markov was
elected to the St. Petersburg Academy of Science in 1886, attaining full
membership in 1896. With A.M. Liapunov, Markov became the most eminent of
Chebyshev's disciples in probability of the Petersburg ``School", and
remained closest to his teacher's ideas. The writings of Markov and Liapunov
placed probability on the level of an exact mathematical science. Markov's
published probabilistic work has in much of its background correspondence with
A.V. Vasiliev, professor at Kazan University, a graduate of the same Petersburg
gymnasium, and also a student of Chebyshev. Indeed several important papers
of Markov, including the one in which ``Markov chains" first appear in Markov's
writings in 1906, were published in the {\it Izvestiia (Bulletin)} of the
Physico-Mathematical Society of Kazan University.
The initial impetus for Markov's work in probability theory was Chebyshev's
proof,
which was incomplete, of the Central Limit Theorem. Bouncing his ideas off
Vasiliev,
Markov begins in 1898 by replacing one of Chebyshev's conditions, while
persisting with
Chebyshev's approach via the method of moments. Liapunov's Theorem on the
Central
Limit problem, published in 1901, differs not only in its approach (by
characteristic
functions, originating with Cauchy and I.V. Sleshinsky (1854-1931)) but also in
its
level of generality. This motivated Markov to wonder whether the method of
moments
might not be suitably adapted to give the same result; and he finally achieved
this
in 1913 in the 3rd edition of his {\it Ischislenie Veroiatnostei (Calculus
of Probabilites)}.
Markov was embroiled in several controversies with the Moscow mathematician P.A.
Nekrasov (1853-1924), one of which led (Seneta, 1984, 1996) to Markov's
outstanding contribution to probability theory, the concept of chain dependence
of random variables. The first of these controversies was initiated by a
probabilistic paper of Nekrasov in 1898, dedicated to Chebyshev (!) and
containing
no proofs. It was followed by about 1,000 pages of obscure and verbose argument
in
{\it Matematicheskii Sbornik}. In its attempt to establish now-standard local
and
global theorems of Central Limit type for large deviations, this work of
Nekrasov
was ahead of its time, but was only partly successful. Its specific
inaccuracies were
criticised by Markov and Liapunov, who never understood the general direction;
the task of so doing was formidable. Moreover, in the course of these writings
in 1902 Nekrasov claimed that pairwise independence of summands was a necessary
condition for the Weak Law of Large Numbers (WLLN) to hold. He had examined,
he said, the ``logical underpinnings" of the way the Bienaym\'{e}-Chebyshev
Inequality was used to prove the WLLN. The {\it observed} stability of
averages in everyday life, through the claimed consequent necessity of
pairwise independence, justified the doctrine of free will. It was this
attempt to use mathematics and statistics in support of theological doctrine
which led Markov to construct a scheme of {\it dependent} random variables in
his Kazan paper, which ends, without ever mentioning Nekrasov explicitly, with
the words
\begin{quote}
``Thus, independence of quantities does not constitute a necessary condition
for existence of the law of large numbers."
\end{quote}
It was in 1902 also that Markov protested over the reversal by the tsar of
election as Honorary Member of the Academy of Science of A.M. Gorky (Peshkov).
Markov refused subsequently to accept any awards (``orders") from the Academy,
or to
act as ``agent of the government" in relation to students at the university.
He came into conflict with the Council of Petersburg University in 1905
about the procedure for relaxing the quota on admission of Jews. In 1912 when
the Synod of the Russian Orthodox Church excommunicated Leo Tolstoy, Markov
likewise requested excommunication. His character and beliefs in combination
with his scientific eminence were very acceptable to the incoming political
system following the October revolution in 1917, and contributed in having the
Petersburg School put into exclusive eminence in Soviet mathematical
historiography (in contrast to the Moscow School, of which Nekrasov and later
Egorov and Luzin were members).
The same historiographic tendencies have progressively ascribed the
Bienaym\'{e}-Chebyshev Inequality and the method of moments to Chebyshev alone.
But just as Chebyshev in the 1874 paper had given Bienaym\'{e} due credit, so
Markov
too was ever a defender of Bienaym\'{e}'s priority. In response to a statement
of
Nekrasov that the idea of Bienaym\'{e} is exhausted in the works of Chebyshev
who, Nekrasov continues, himself had remarked on this in 1874, Markov in 1912
writes (characteristically)
\begin{quote}
``The reference here to Chebyshev is misleading, and the statement of P.A.
Nekrasov that the idea of Bienaym\'{e} is exhausted is contradicted by a
sequence of my papers containing a generalization of the method of
Bienaym\'{e} to settings which are not even touched on in the writings of
P.A. Nekrasov."
\end{quote}
\noindent The first of these papers which he lists is the Kazan paper of 1906,
written to contradict Nekrasov's assertions about the necessity of pairwise
independence for the WLLN.
Markov retired from the university in 1905, but continued to teach probability
theory there. From 1904 to 1915 he wrote letters to newspapers on current
social issues, and especially on education (Sheynin, 1989); the press coined
for him the name {\it Neistovy Andrei (Andrew the Furious)}. In 1915 he
opposed the programme proposed by P.S. Florov and Markov's continuing b\^{e}te
noir Nekrasov about changes to the school mathematics syllabus. There are
good biographies of Markov, most notably by his son (Markov, 1951) and
Grodzensky (1987).
It is, however, his views of and contributions to statistics which deserve to
be addressed also.
On his retirement from the university, continuing to seek practical applications
of probability theory, he participated from the beginning in deliberations on
running the retirement fund of the Ministry of Justice, following in the
footsteps
of his probabilistic predecessors V. Ya. Buniakovsky, M.V. Ostrogradsky and
Bienaym\'{e}.
Markov's attention was turned to mathematical statistics through his
correspondence (Ondar, 1981) with Chuprov (q.v.) which begins 2 November 1910 with a
postcard to the latter criticizing him for mentioning Nekrasov's name in the
same breath as Chebyshev's, in Chuprov's erudite {\it Ocherki po Teorii
Statistiki (Topics in the Theory of Statistics)} of 1909, which had just come to
Markov's attention. From such inauspicious beginnings, in which Markov,
claiming to judge all work only from a strictly mathematical point of view,
dismissed the work of Karl Pearson (q.v.) amongst others, grew a lively correspondence
on
the topic of dispersion theory. At the same time as the interests of the
statistician
Chuprov were turned progressively to a mathematical direction, Markov's
negative attitude to statistics softened, and in the end, out of the
correspondence came elegant and important theoretical contributions from both
(Heyde and Seneta, 1977, Section 3.4). Indeed, the correspondence marks the
coming together in the Russian Empire of probability and statistics into
mathematical statistics. The correspondence ends in early 1917. In the course
of it, Markov was led in 1913 to modelling the alternation of vowels and
consonants in several Russian literary works by a two-state Markov chain and
estimation in the model using dispersion-theoretic ideas.
Markov was also interested, through the influence of Chebyshev, in the
classical linear model which he treated in his {\it Ischislenie Veroiatnostei}
in various editions. The inappropriate name ``Gauss-Markov theorem" seems
ultimately to arise from these treatments.
Markov's Inequality is the name given to the result $P(Y \geq a) \leq EY/a$
where \ $Y$ \ is a non-negative random variable and $a > 0$ . It
appeared in the 1913 edition of Markov's {\it Ischislenie Veroiatnostei}, and
is more fundamental than the Bienaym\'{e}-Chebyshev Inequality, although the
simple proof used by Bienaym\'{e} can be modified to prove it also.
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\begin{thebibliography}{3}
\bibitem{1} Grodzensky, S. Ya. (1987). {\it Andrei Andreevich Markov. 1856-1922}.
Nauka, Moscow.
\bibitem{2} Heyde, C.C. and Seneta, E. (1977). {\it I.J. Bienaym\'{e}. Statistical
Theory Anticipated}
Springer, New York.
\bibitem{3} Markov, A.A. (1951). {\it Izbrannie Trudy. (Selected Works).} ANSSSR,
Leningrad.
\bibitem{4} Ondar, Kh.O. (1981). {\it The Correspondence Between A.A. Markov and
A.A. Chuprov
on the Theory of Probability and Mathematical
Statistics.}
Springer, New York
[Transl. by C. and M. Stein].
\bibitem{5} Seneta, E. (1984). The central limit problem and linear least squares
in pre-revolutionary Russia. The background.
{\it Mathematical Scientist}, {\bf 9}, 37-77.
\bibitem{6} Seneta, E. (1996). Markov and the birth of chain dependence theory.
{\it International Statistical Review}, {\bf 64},
255-263.
\bibitem{7} Sheynin, O.B. (1989). A.A. Markov's work on probability. {\it
Archive for History of Exact Sciences},
{\bf 39}, 337-377.
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\hfill{E. Seneta}
\end{thebibliography}
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