{"id":21437,"date":"2025-03-07T17:14:02","date_gmt":"2025-03-07T16:14:02","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=21437"},"modified":"2025-03-10T10:25:02","modified_gmt":"2025-03-10T09:25:02","slug":"ivan-nikolaevich-sanov","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2025\/03\/07\/ivan-nikolaevich-sanov\/","title":{"rendered":"Ivan Nikolaevich Sanov (1919 - 1968)"},"content":{"rendered":"
\"Painting<\/a>
Spring Day, by Nikolai Matveevich Pozdneev, 1959<\/figcaption><\/figure>\n

Probability.<\/strong> 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 $B\\subset\\mathcal{P}(E)\\subset\\mathbb{R}^{|E|}$,
\n\\[
\n-\\inf_{\\mathrm{interior}(B)}\\mathrm{H}(\\cdot\\mid\\mu)
\n\\leq\\varliminf_{n\\to\\infty}\\frac{\\log\\mathbb{P}(L_n\\in B)}{n}
\n\\leq\\varlimsup_{n\\to\\infty}\\frac{\\log\\mathbb{P}(L_n\\in B)}{n}
\n\\leq-\\inf_{\\mathrm{closure}(B)}\\mathrm{H}(\\cdot\\mid\\mu)
\n\\]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 :
\n\\[
\n\\mathbb{P}(L_n\\in B)
\n=
\n\\exp\\Bigr(-n\\inf_B\\mathrm{H}(\\cdot\\mid\\mu)+o(n)\\Bigr).
\n\\]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 $\\nu=\\frac{1}{n}\\sum_k\\delta_{x_k}=L_n(x_1,\\ldots,x_n)\\in\\mathcal{P}(E)$,
\n\\[
\n\\mathbb{P}(X_1=x_1,\\ldots,X_n=x_n)
\n=\\prod_i\\mu_i^{n\\nu_i}
\n=\\exp\\Bigr(-n\\Bigr(\\mathrm{S}(\\nu)+\\mathrm{H}(\\nu\\mid\\mu)\\Bigr)\\Bigr)
\n\\]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 Cram\u00e9r theorem obtained by Harald Cram\u00e9r (1893 - 1985). It constitutes nowadays one of the important basic elements of modern Large Deviations Theory, developped notably by S. R. Srinivasa Varadhan (1940 - ).<\/p>\n

Algebra.<\/strong> 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!<\/p>\n

Sanov studied algebra as well as probability, and it seems that it is the same for Burnside.<\/p>\n

Further reading.<\/strong><\/p>\n