Bernstein, Sergei Natanovich

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This article Sergei Natanovich Bernstein was adapted from an original article by E. Seneta, which appeared in StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. The original article ([ StatProb Source], Local Files: pdf | tex) is copyrighted by the author(s), the article has been donated to Encyclopedia of Mathematics, and its further issues are under Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the Category StatProb.

Sergei Natanovich BERNSTEIN

b. 22 February 1880 (o.s.) - d. 26 October 1968

Summary Bernstein's training in and continuing contacts with Paris led him to combine analytical writing with the traditions of the St.Petersburg School in probability. Martingale differences appear in his work, and best known are his extensions of the Central Limit Theorem to weakly dependent random variables.

Bernstein was born in Odessa in the then Russian Empire. His father was a doctor and university lecturer, and the family was Jewish, with the attendant difficulties. On completing high school Bernstein went to Paris for his mathematical education, and defended a doctoral dissertation in pure mathematics in 1904 at the Sorbonne. He returned in 1905 and taught at Kharkov University from 1908 to 1933; the system of czarist universities, and possibly his Jewishness, made it necessary to defend another doctoral dissertation in pure mathematics in 1913. Jerzy Neyman (q.v.) recollects lectures at Kharkov University in 1915 or 1916 by Bernstein on probability, and that it was Bernstein who suggested to him that he read Karl Pearson's Grammar of Science. Clearly, given Neyman's influence on the direction of mathematical statistics subsequently, Bernstein was influential in this sense also, quite apart from his many and striking contributions to probability theory.

After the revolution Bernstein became professor at Kharkov University, and became active in the Soviet reorganization of tertiary institutions, as a national commissar for education, for example in the establishment in Kharkov of the All-Ukrainian Scientific Research Institute of Mathematical Sciences in 1928. During the quickly-suppressed period of Ukrainianization within the time when Kharkov (Kharkiv in Ukrainian) was capital (1919-1934) of the Ukrainian SRS, he refused to use the Ukrainian language, although there is a publication of his in 1928 (on the concept of correlation between statistical variables) written in this language. This would have put him at variance with the chairman (1927-1933) of national commissars for education of Ukraine, M.V. Skrypnyk (1872-1933, who under Stalinist pressure committed suicide), and mathematicians such as M.P. Kravchuk (q.v.).

Possibly finding Markov's (q.v.) Ischislenie Veroiatnostei dated as a didactic aid, Bernstein produced an elegant textbook Teoriia Veroiatnostei which first appeared in 1927, went to 2nd and 3rd editions in 1934, with a final 4th edition in 1946. There were substantial changes in the successive editions, and the 4th edition contains a significant amount of new research material, especially on his own incomplete axiomatization of probability, and on inhomogeneous Markov chains. He was very familiar with the probabilistic work of the Petersburg School and wrote a splendid commentary on Chebyshev's (q.v.) probabilistic work in 1945 and can well be thought of as succeeding Liapunov (who left it in 1902) at Kharkov University. Even though the origins of the Petersburg direction themselves were largely under French influence due to Buniakovsky and Chebyshev, Bernstein's own training in and contact with Paris, shown in his heavily analytical writing, helped him combine its manifestation with then-current European thinking. The scope of his probabilistic work in general was ahead of its time, and his writings, including his book, helped significantly to shape the development of probability, and not only in the USSR.

Berstein took a keen interest in the methodology of teaching mathematics at secondary and tertiary levels, and popularizing its use. His official bibliography of about 265 items contains numerous book reviews and articles in Pedagogicheskii Sbornik in prerevolutionary years, and after in journals such as Nauka na Ukraine, for example an article in 1922 entitled "On the application of mathematics to biology". These activities doubtless contributed to his appointment as a national commissar for education, and, at least in the years prior to 1933, helped to further the standing of mathematics.

From 1933 Bernstein worked at the Mathematical Institute of the USSR Academy of Sciences in Leningrad (now again called St. Petersburg), and also taught at the University and Polytechnic Institute. From January, 1939, Bernstein worked also at Moscow University. He and his wife were evacuated to Kazakhstan before Leningrad was blockaded by German Armies from September 8, 1941 to January, 1943. From 1943 he worked at the Mathematical Institute in Moscow.

In the years 1952-1964 he spent much time in the editing and publication of the four-volume collection of his mathematical works, which contains commentaries by his students and experts in various fields. The first 3 volumes deal with essentially non-probabilistic themes. The 4th volume is entitled "Theory of Probability and Mathematical Statistics [1911-1946]. One problem to which he kept returning was the accuracy of the normal approximation to the normal distribution. In fact a theme of his work was reexamination in a new light of the main existing theorems of probability theory, such as extension to dependent random variables of the Weak Law of Large Numbers. (This law deals with conditions under which the sample means ${\bar X}_n = (X_1 + X_2 + \cdots + X_{n})/n$ formed from a sequence $\{X_{n}\}$ of random variables converge in probability to a constant, as $n$ increases.) The characterization of the normal distribution through independence of linear forms in two random variables is usually referred to as Bernstein's Theorem. The name Bernstein's Inequality has its origin in a paper of 1924 and is applied to a number of inequalities, the most common of which is $P(X \geq a) \leq e^{-at} M(t)$, for $t > 0$, where $ M(t)=E(e^{Xt}) $. (This follows immediately from Markov's Inequality.)

Little known (although partly translated into English) is a surprisingly advanced (for its time, 1924) mathematical investigation in population genetics, involving a synthesis of Mendelian inheritance and Galtonian "laws" of inheritance.

The idea of martingale differences appears in his work; and probably best-known are his extensions of the Central Limit Theorem to ``weakly dependent random variables". The classical limit theorems (the Weak Law of Large Numbers and the Central Limit Theorem) are concerned with the probabilistic behaviour as $n \rightarrow \infty$ of the partial sums $\{S_{n}\}$ where $ S_n=X_1+X_2+...+X_n $ of a sequence of $\{X_{n}\}$ of "independent random variables with zero mean $(E X_n = 0)$. For the more general concept of a martingale difference sequence of random variables $\{X_{n}\}$ , the property $E(X_n | X_{n-1} , X_{n-2} , \cdots, X_1) = 0$ is retained. These are Bernstein's "first order dependent random variables". Since $X_n = S_n - S_{n-1}$ , this defining property can be formulated as $E(S_n - S_{n-1} | S_{n-1} , S_{n-2}, \cdots, S_1) = 0$. The sequence $\{S_{n}\}$ of partial sums is now called a "martingale. It has the property $E(S_n | S_{n-1} , \cdots, S_1) = S_{n-1}$ and the sequence $\{X_{n}\}$ is thus one of "martingale differences".

For the statistician especially, of interest is a paper of 1941 entitled "On the "fiducial" probabilities of Fisher."

The Bernstein polynomials have a number of uses in probabilistic contexts.

Bernstein's students included G.A. Ambartsumian, V.P. Savkevich, O.V. Sarmanov, H.A. Sapogov. An epitaph which he might have chosen for himself preceded a prize-winning work of his (1911):

La vie est brève

Un peu de rêve

Un peu d'espoir

Et puis bonsoir.


[1] Bernstein, S.N. (1964). Sobranie Sochinenii (Collected Works, 4 vols.) Gostehizdat, Moscow-Leningrad.
[2] Bogoliubov,A.N. (1997). Serhiy Natanovych Bernshtein (1880-1968). In: Instytut Matematyky. Narysy Istorii. 17, 175-189. [In Ukrainian.] Published by: Instytut Matematyky Ukr. AN, Kyiv.
[3] Kolmogorov, A.N. and Sarmanov, O.V. (1960). On the writings of S.N. Bernstein on the theory of probabilities. [in Russian]. Teoriia Veroiatnostei i ee Primeneniia, 5, 215-221.
[4] Reid, C. (1982). Neyman - from life. Springer, New York.
[5] Seneta, E. (1982). Bernstein, Sergei Natanovich. Encyclopedia of Statistical Sciences (S. Kotz and N.L. Johnson, eds.) Wiley, New York 1, 221-223.

Reprinted with permission from Christopher Charles Heyde and Eugene William Seneta (Editors), Statisticians of the Centuries, Springer-Verlag Inc., New York, USA.

How to Cite This Entry:
Bernstein, Sergei Natanovich. Encyclopedia of Mathematics. URL:,_Sergei_Natanovich&oldid=39176