Probability. One of my favorite theorems in probability theory is the following refined law of large numbers known as the Sanov large deviations principle : if we consider the empirical measure $L_n=L_n(X_1,\ldots,X_n)=\frac{1}{n}\sum_{k=1}^n\delta_{X_k}$ made with independent and identically distributed random variables $X_1,\ldots,X_n$ of law $\mu$ on a finite set $E$, then for all $B\subset\mathcal{P}(E)\subset\mathbb{R}^{|E|}$,
\[
-\inf_{\mathrm{interior}(B)}\mathrm{H}(\cdot\mid\mu)
\leq\varliminf_{n\to\infty}\frac{\log\mathbb{P}(L_n\in B)}{n}
\leq\varlimsup_{n\to\infty}\frac{\log\mathbb{P}(L_n\in B)}{n}
\leq-\inf_{\mathrm{closure}(B)}\mathrm{H}(\cdot\mid\mu)
\]where $\mathrm{H}(\nu\mid\mu)=\sum_i\nu_i\log\frac{\nu_i}{\mu_i}$ is the Kullback-Leibler divergence. When $B$ is sufficiently regular, this gives a Boltzmann-Gibbs approximation of the law of the empirical measure :
\[
\mathbb{P}(L_n\in B)
=
\exp\Bigr(-n\inf_B\mathrm{H}(\cdot\mid\mu)+o(n)\Bigr).
\]It is meaningful in statistical physics and information theory, and differs from the Central Limit Theorem approximation. Moreover, this large deviation principle extends, by discretization, to any Polish space $E$ with the topology of weak convergence on $\mathcal{P}(E)$. The proof relies on the fact that for all $(x_1,\ldots,x_n)\in E^n$, denoting $\nu=\frac{1}{n}\sum_k\delta_{x_k}=L_n(x_1,\ldots,x_n)\in\mathcal{P}(E)$,
\[
\mathbb{P}(X_1=x_1,\ldots,X_n=x_n)
=\prod_i\mu_i^{n\nu_i}
=\exp\Bigr(-n\Bigr(\mathrm{S}(\nu)+\mathrm{H}(\nu\mid\mu)\Bigr)\Bigr)
\]where $\mathrm{S}(\nu)=-\sum_i\nu_i\log \nu_i$ is the Boltzmann-Shannon entropy of $\nu$, and on the quantitative Stirling formula $|\{x\in E^n:L_n(x)=\nu\}|=\exp(n\mathrm{S}(\nu)+o(n))$ for all $\nu\in L_n(E^n)$. This theorem was published in 1957 by Ivan Nikolaevich Sanov (1919 - 1968), a soviet mathematician. It can be seen as a sort of dual of the Cramér theorem obtained by Harald Cramér (1893 - 1985). It constitutes nowadays one of the important basic elements of modern Large Deviations Theory, developped notably by S. R. Srinivasa Varadhan (1940 - ).
Algebra. Sanov is also famous for his work on algebra, notably on the Burnside problem, formulated in 1902 by William Burnside (1852 - 1927) : whether a finitely generated group in which all elements have finite order must be a finite group. In particular, Sanov proved in 1940 that the Burnside group $B(r,n)$, which is the group with $r$ generators and all elements of order $n$ (and no more relations), is finite for all $r$ and $n=4$. He also worked on free subgroups of $\mathrm{SL}_2(\mathbb{Z} )$, and proved in 1947 that the subgroup of $\mathrm{SL}_2(\mathbb{Z})$ generated by $$\begin{pmatrix} 1 & 2\\0 & 1\end{pmatrix}\quad\text{and}\quad \begin{pmatrix}1 & 0\\2 & 1\end{pmatrix}$$ is free, and that it is formed by all the matrices of unit determinant of the form $$\begin{pmatrix} 1+4n_1 & 2n_2\\ 2n_3 & 1+4n_4\end{pmatrix}$$for arbitrary integers $n_i$. The freeness can be proved by using the ping-pong or table-tennis lemma of Felix Klein. The formula implies that the membership problem for this subgroup is solvable in constant time. The subject is also related to the famous Tits alternative, formulated by Jacques Tits (1930 - 2021) in 1972 : for all finitely generated linear group over a field, either there exists a solvable subgroup of finite index, or a subgroup isomorphic to a free group on two generators (thus non abelian). The subgroup membership problem, also known as the generalized word problem, is fascinating. It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group for which the word problem is undecidable. Considerable progresses were made on the Burnside problem, notably by Novikov, and counter examples were discovered, such as the Tarski Monsters, while it is still unknown if $B(2,5)$ is finite or not!
Sanov studied algebra as well as probability, and it seems that it is the same for Burnside.
Further reading.
- Imre Csiszár
A simple proof of Sanov's theorem
Bull. Braz. Math. Soc. 37 (2006), no. 4, 453–459 - Raphaël Cerf
On Cramér's theory in infinite dimensions
Panor. Synthèses 23 Société Mathématique de France, Paris, 2007 - Michael Vaughan-Lee
The restricted Burnside problem
Oxford University Press (1993) - Pierre de la Harpe
Topics in Geometric Group Theory
The University of Chicago Press (2000) - William Burnside
Theory of Probability
Cambridge University Press, 1936 - A. A. Borovkov, P. N. Golovanov, V. Ya. Kozlov, A. I. Kostrikin, Yu. V. Linnik, P. S. Novikov, D. K. Faddeev, N. N. Chentsov. Translated by H. Freedman.
Ivan Nikolaevich Sanov (obituary)
Russian Math. Surveys, 24:4 (1969), 159–161
Sanov died at the age of 49, and published only 11 articles. Here is the OCR of his obituary:
A.A. Borovkov, P.N. Golovanov, V.Ya. Kozlov, A.L. Kostrikin, Yu. Vv. Linnik, P.S. Novikov, D.K. Faddeev, N.N. Chentsov,
Russian Mathematical Surveys, 1969, Volume 24, Issue 4, 159-161.
Translated by H. Freedman.IVAN NIKOLAEVICH SANOV Obituary
The gifted Soviet mathematician Ivan Nikolaevich Sanov died after a
grave illness on September 7, 1968.I.N. Sanov was born on April 12, 1919 in Meshchovsk in the Province
of Kaluzhsk. In 1929 his family moved to Leningrad. Sanov completed his
secondary education at School 107 in Leningrad. Even while still at school
he showed outstanding ability in mathematics. In his pre-final year at
school he took part in the first Mathematical Olympiad in Leningrad (based
on the syllabus of the final class) and was among the winners, repeating
his success the following year. In 1935 he entered the Mathematics-
Mechanics Faculty of the University of Leningrad, from which he graduated
in 1940. After graduation he worked for one year as an Assistant Lecturer
at the Department of Algebra at the Pedagogical Institute, at the same
time doing research as an external student at the University of Leningrad.While still a student he produced the remarkable paper “Solution of
Burnside’ s problem for exponent 4” [1]. Until then Burnside’s famous
problem had been solved for exponent 3 only.Sanov was in active service with the army throughout the whole Second
World War. Commanding at first a platoon and then a company of an anti-
aircraft artillery regiment, he took part in the battles on the fronts of
Leningrad, the South-West and the Ukraine. He then went on with the Soviet
army on its missions of liberation to Rumania, Poland and Germany.For his war service I.N. Sanov was decorated with the “Order of the
Red Star”, and with the medals “For the Defence of Leningrad”, “For
the Liberation of Warsaw”, ‘For the Capture of Berlin” and “ For Victory
over Germany in the ar 1941-1945”.In 1943 Sanov joined the ranks of the Communist party.
After his demobilization (in 1946) he worked at the University of
Leningrad and (from 1948) at the Leningrad branch of the Mathematical
Institute.Already before the war Sanov had started to investigate the Burnside
problem by means of the apparatus of Lie’s rings; after his demobilization
he returned to these investigations, which were summed up in two long papers
[6], [7] in Izvestia Akad. Nauk SSSR of 1951 and 1952. There he developed
an original technique directed towards obtaining non-trivial relations in
groups satisfying Burnside’s condition. The application of this technique
enabled him, in particular, to refine a result of O. Grün on the order of
a two generators group of exponent 5.Sanov’s work had a strong influence on further research on the
restricted Burnside problem (on the bounds of the orders of finite groups
with a prescribed number of generators and a prescribed exponent) and
references to his paper appear in the literature right up to the present
time.Among his other results in algebra we should mention a very simple
matrix representation for free groups [3].In his paper “On functions with integral parameters and of least
deviation from zero” [5] an elegant geometric solution is given of the
problem (investigated earlier by M. Fekete) of the least deviation from
zero of generalized polynomials with integral coefficients. Sanov’s
bounds contain more precise constants (factors) than those of Fekete and,
in addition, he shows that these bounds cannot be improved without
additional restrictions.In the autumn of 1949 Sanov was assigned for one year to the Korean
People’s Democratic Republic to help organize mathematical education in the
University of Pyongyan, where he worked as an adviser to the Dean of the
Physics-Mathematics Faculty and was Head of the Department of Higher
Mathematics.In 1952 Sanov came to work in Moscow. Here he began research on some
problems in the theory of probability and mathematical statistics. In
these fields, new to him, he obtained a number of deep results. Among
these we should mention, in the first instance, the interesting approach
to the investigation of problems of large deviations of sums of
equidistributed random variables [9], which is connected with the concept
of information distance in the space of distribution functions.In this paper he discusses a generalization of results obtained, for
discrete random variables, to the case of arbitrary random variables with
a fixed distribution function.In 1961 at the IV All-Union Conference Sanov delivered a paper on a
new method of computing the asymptotic probability of large deviations of
a sequence of random quantities, which aroused great interest among the
experts.From among his works in algebra in this period we mention two
publications in the Siberian Mathematical Journal, [10] and [11]. In the
first paper he gives explicit formulae for the coefficients of a quadratic
form after eliminating a number of variables, satisfying one or several
relations, as well as for arbitrary minors of the symmetric matrix of this
form. In the second paper he investigates Euclid’s algorithm and one-sided
factorization into prime factors for matrix rings.For his successful solutions of a number of problems of practical
importance by a Decree of the Presidium of the Supreme Soviet, Sanov was
awarded the Order of Lenin and several medals.I.N. Sanov was a typical representative of the Leningrad mathematical
school, In the best tradition of this school, he devoted much of his
energy and attention to training highly qualified experts in algebra and
probability.Death caught him in the prime of his life; he developed many new ideas
and left behind a number of unpublished papers. The untimely death of
Ivan Nikolaevich Sanov is a great loss for Soviet Science.LIST OF SANOV'S PUBLISHED PAPERS
1940
[1] Solution of Burnside’s problem for exponent 4, Uchen. zap. Leningrad Univ. Ser. Fiz, Mat. 10, 166-170.1946
[2] Periodic groups with small periods, Ph.D. dissertation, Leningrad University,1947
[3] A property of a representation of a free group, Dokl. Akad. Nauk SSSR 57, 657-659.
[4] On Burnside’s problem, Dokl. Akad. Nauk SSSR 57, 759-761.1949
[5] On functions with integral parameters and of least deviation from zero, Uchen. zap. Leningrad Univ. Ser, Fiz-Mat. II, 32-46.1951
[6] On a certain system of relations in periodic groups with period a power of a prime number, Izv. Akad. Nauk SSSR, Ser. Mat. 15, 477-502.1952
[7] A connection between periodic groups of prime period and Lie rings, Izv. Akad. Nauk SSSR, Ser. Mat. 16, 23-58.
[8] A new proof of Minkowski's theorem, Izv, Akad, Nauk SSSR, Ser. Mat. 16, 101-112.1957
[9] On the probability of large deviations of random quantities Mat. Sb. 52, 11-44, Amer, Math, Soc. Transl. (2) |, 213-244.1967
[10] On an elimination formula, Sibirsk. Mat. Zh. 8, 841-845.
[11] Euclid's algorithm and one-sided factorization for matrix rings, Sibirsk. Mat. Zh, 8, 846-852.
