Mikhail Iosifovich (Misha) Gordin died on March 17th, 2015, in St. Petersburg, Russia, after a long illness. A senior researcher at the Steklov Institute of Mathematics of the Russian Academy of Sciences in St. Petersburg (POMI), he was also a faculty member at the St. Petersburg State University where he taught classes in probability theory.
Among many signs of professional recognition, M.I. was a Taft
Fellow at the University of Cincinnati in 2009, and a Gauß
Professor in Göttingen in 2014. He played a vital role in the
mathematical community of St. Petersburg; in particular, he was
a board member of the St. Petersburg Mathematical Society. A
volume in honor of M.I. Gordin's 70th birthday was published in
the Zapiski Nauchnykh Seminarov Sankt-Peterburgskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Rossiiskoi
Akademii Nauk (Vol. 431 of the Notes of Scientific Seminars of
the St. Petersburg Department of the Steklov Mathematical
Institute, Russian Academy of Sciences). This volume will be
translated into English and published by Springer in the Journal
of
Mathematical Sciences.
Among many important mathematical results obtained by M.I., his
1969 paper "The central limit theorem for stationary processes"
was a seminal contribution to the study of the Central Limit
Theorem (CLT) for partial sums of dependent processes, that has
influenced a large number of mathematicians to this day. The
method of the martingale difference sequence approximation
introduced in this paper has proved very useful in many fields
besides probability theory, in particular, in dynamical systems.
Misha Gordin was born in Leningrad, the Soviet Union, on
September 9th of 1944, to Iosif Mikhailvich (Osya) Gordin and
Rosa Iudovna Welitkranz. His father was originally from a
Siberian city of Kansk, and his mother came from Poland in 1932.
She was a medical doctor, while I.M. held leading positions in
various organizations responsible for hydro- and irrigation
systems in Leningrad and the Leningrad region.
Misha grew up a precocious child who is still remembered by his
classmates as an exceptionally bright, self-motivated learner.
He participated in the mathematical circles, and later taught
there himself. In 1961 he started studying mathematics at the
Leningrad State University, where he eventually became an
advisee of I.A. Ibragimov in the field of probability theory. By
the time he defended his PhD (Candidate of Sciences) thesis in
1970, the situation in the Soviet Union has changed.
Institutional anti-semitism crept in and, as a result, M.I.
could not find any job for several months. Eventually he was
employed by the All-Union Research Institute for Electrical
Measurements (VNIIEP, now Electromera) in Leningrad. He worked
there for more than 20 years, and even though many (including
his co-workers) felt that he should have been employed at a more
research-oriented organization, he still found some of the
problems at the VNIIEP inspiring. As a result, he obtained
several results in logic and, moreover, he supervised research
of several younger mathematicians in the field. Unfortunately,
his results in logic have not been published. In particular, at
that time he had his first official PhD student, Alex Wolpert,
who is now a Professor at the Department of Computer Science,
Roosevelt University, Chicago, USA.
Even though his situation prevented him from assuming an
official role of an advisor, many (former) Leningrad
mathematicians acknowledge significant influence on their
development as mathematicians; among them, Sergey Fomin
(University of Michigan) and Vadim Kaimanovich (University of
Ottawa).
Finally, with the changes that came to the Soviet Union in the 90s, M.I. moved to POMI, where he worked for more than 20 years. This also brought an opportunity to travel abroad, which was impossible during the Soviet times. His first foreign trips were to Prague and Göttingen, which were always the special places he liked to visit in later years as well. Moreover, in Göttingen M.I. supervised (jointly with M. Denker) another PhD dissertation, by Rada Dakovic (Matic).
These political changes also allowed him to communicate and
collaborate freely with a larger mathematical community
including Rick Bradley, Dalibor Volny, and his collaborators
Manfred Denker, Friedrich Götze, Magda Peligrad, and Michel
Weber.
M.I. chose to remain in St. Petersburg even as many
mathematicians left the country, including members of his
immediate family. His many interests besides mathematics
included music (especially jazz), art and literature. M.I.
Gordin is survived by his wife, Natalia Davydovna (Natasha)
Gordina, his daughter, Masha Gordina, his granddaughter, Mira,
and his son-in-law, Sasha Teplyaev.
Revised: April 2015, based on the 2011 version
1970: Ph.D. Leningrad State University
Dissertation: Some results in the theory of stationary
random processes
Advisor: I. A. Ibragimov
1966: M. Sc. (Diploma) Leningrad State University
1992-2015: Senior Researcher, V.A. Steklov Institute of
Mathematics at St. Petersburg (POMI), Russian Academy of
Sciences
1970-1992: Leading researcher, Senior Researcher, Junior
Researcher, at the All-Union Research Institute for Electrical
Measurements (now Electromera), St. Petersburg, Russia
2014: Gauß-Professur, Akademie der Wissenschaften zu Göttingen
2009: Taft Fellow, University of Cincinnati
2006: POMI Award for the Best Scientific Paper of the Year
2004: POMI Award for the Best Scientific Paper of the Year
2014: University of Göttingen, Germany
2012: Pennsylvania State University, USA
2009: University of Cincinnati, Taft Research Fellowship, USA
2003: University of Strasbourg, France
2001: University of Strasbourg, France
2000: University of Strasbourg, France
1999: University Paris-Sud, France
1996: University Lille-1, France
1995: Institute for Dynamical systems, University of Bremen,
Germany
1993: University Paris-Sud, France
1990: Department of Mathematics, Charles University, Prague,
Czechoslovakia
2009-2011, 2004-2006, 1999-2002, 1996-1997: joint research
grants of the Deutsche Forschungsgemeinschaft (DFG) and the
Russian Foundation for Basic Research (RFBR)
2010-2012, 2005-2007, 2002-2004, 1999-2001, 1996-1998,
1993-1995:
research grants of the RFBR
2003, 1999: travel grants of the RFBR
1999, 1998: joint research grants of the CNRS (France) and the
Russian
Academy of Sciences
1997-2007: research and travel grants of INTAS (International
Association for the promotion of co-operation with scientists
from the New Independent States of the Former Soviet Union)
Referee for Probability Theory and its Applications, Annals of Institute Henry Poincare, Stochastic Processes and their Applications, Stochastics and Dynamics, Functional Analysis and its Applications, Saint Petersburg Mathematical Journal, Nonlinearity, Zapiski nauchnykh seminarov POMI (translated in Journal of Mathematical Sciences), Israel Science Foundation Proposal, Ph.D. dissertations in Russia, France and Germany
i) Limit theorems for weakly dependent variables: in
particular, for ran- dom processes produced from hyperbolic or
partially hyperbolic dynamical systems; the main ambition here
is to avoid using such non-canonical tools as partitions (Markov
ones or similar) basing instead on internal structures discussed
in the next point;
ii) Study of internal structures related to hyperbolic and
weakly hyperbolic dynamics, in particular, synchronous and
asynchronous homoclinic equiv- alence relations and groupoids
related to strictly hyperbolic discrete time dynamical systems;
these structures, when we are given a dynamically invari- ant
Gibbs measure, provide a partial substitute for systems of
sigma-fields which are more familiar to the probabilists;
iii) Limit distribution of the spectra of large random matrices,
in particu-
lar, asymptotics of correlation functions and explanation of the
appearance of determinantal processes in the limit; exact
probabilistic models of free probability (of which random
matrices give, according to D. Voiculescu, an asymptotical probabilistic
model).
2011: Combinatorial, bialgebraic and analytic aspects of free
probability
(Vienna, Austria)
Ergodic theorems, group actions and applications (Eilat, Israel)
Dynamical Systems (Göttingen, Germany)
2010: Proprietes stochastiques des systemes dynamiques et
marches aleatoires
(Roscof, France)
Limit Theorems for Dependent Data and Applications (Paris)
Vilnius Conference on Probability and Mathematical Statistics
(Vilnius, Lithuania)
Free Probability and Random Matrices (Bielefeld, Germany)
2007: Workshop on Asymptotic Statistics and Its Applications in
Honor of
Ildar I. Ibragimov (Bordeaux, France)
Ergodic Theory and Limit Theorems (Rouen, France)
Conformal Structures and Related Problems (Goettingen, Germany)
Asymptotic Statistics and Its Applications (Bordeaux, France)
2006: Stochastic Processes and Random Fractals (Lille, France)
European Meeting of Statisticians (Torun, Poland)
Prague Conference on Probability, Statistics and Random
Processes (Prague, Czech Republic)
2005: Analytic Methods in Number Theory, Probability and
Mathematical
Statistics, Yu.V. Linnik's Memorial (Saint Petersburg, Russia)
Quantum Chaos and Random Matrices (Bielefeld, Germany)
[1] M. Denker, M. Gordin. Limit theorems for von Mises statistics of a measure preserving transformation. Probab. Theory Related Fields
[2] M. Gordin. CLT for stationary normal Markov
chains via generalized coboundaries. Limit theorems in
probability, statistics and number theory, 93-112, Springer
Proc. Math. Stat., 42, Springer, Heidelberg,
2013.
[3] M. Denker, M. Gordin. The Poisson limit for
automorphisms of two-dimensional tori defined by continued
fractions. (Russian) Zap. Nauchn. Sem. S.-Peterburg.
Otdel. Mat. Inst. Steklov. (POMI) 408
(2012), Veroyatnost i Statistika. 18, 131-153, 325-326;
translation in
J. Math. Sci. (N. Y.) 199 (2014), 139-149.
[4] M. Gordin, M. Peligrad. On the functional CLT
via martingale approximation. Bernoulli, vol. 17 (2011),
424-440.
[5] M.I. Gordin. Homoclinic processes and invariant
measures for hyperbolic automorphisms of tori, Zap.
Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI),
vol. 368 (2009), P. 501-505 (Russian); transl. in J.
Math. Sci. (N. Y.) 167 (2010), 501-505.
[6] M.I. Gordin. Martingale-co-boundary
representation for a class of stationary random fields,
Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.
(POMI), vol. 364 (2009), P. 88-108 (Russian); transl.
in J. Math. Sci. (N. Y.) 163 (2009), no. 4, 363-374
[7] Götze, Friedrich; Gordin, Mikhail. Limit
correlation functions for fixed trace random matrix ensembles.
Comm. Math. Phys. 281 (2008), no. 1, P. 203-229.
[8] Götze, F. ; Gordin, M. I. ; Levina, A. The limit
behavior at zero of correlation functions of random matrices
with a fixed trace. Zap. Nauchn. Sem. S.-Peterburg.
Otdel. Mat. Inst. Steklov. (POMI) 341 (2007), Veroyatn.
i Stat. 11, 68-80, 230, Russian; transl. in J. Math.
Sci. (N. Y.) 147 (2007), no. 4, P. 6884-6890.
[9] Mikhail Gordin, Michel Weber. A borderline
Gaussian random Fourier series for the sample convergence in
variation. J. Math. Anal. Appl.,
[10] Mikhail Gordin. A remark on the martingale
method for proving the central limit theorem for stationary
sequences. Zap. Nauchn. Sem. S.Peterburg. Otdel. Mat.
Inst. Steklov. (POMI), 311 (2004) Veroyatn. i Stat. 7,
124-132 (Russian); transl. in J. Math. Sci. (N. Y.) 133 (2006),
no. 3, 1277-1281.
[11] Mikhail Gordin, Hajo Holzmann. The central
limit theorem for stationary Markov chains under invariant
splittings. Stoch. Dyn. 4 (2004): 1, 15-30.
[12] Manfred Denker, Mikhail Gordin, Anastasiya
Sharova. A Poisson limit theorem for hyperbolic toral
automorphisms. Illinois J. Math. 48 (2004): 1,
1-20.
[13] Friedrich Götze, Mikhail Gordin. Limiting
distributions of theta series on Siegel half-spaces.
Algebra i Analiz 15 (2003): 1, 118-147; reproduced in
St. Petersburg Math. J. 15 (2004): 1, 81-102.
[14] Manfred Denker, Mikhail Gordin. On conditional
central limit theorems for stationary processes. Probability,
statistics and their applications: papers in honor of Rabi
Bhattacharya; IMS Lecture Notes Monogr. Ser., vol. 41 (2003),
133-151.
[15] Manfred Denker, Mikhail Gordin, Stefan-M.T
Heinemann. On the relative variational principle for fibre
expanding maps. Ergodic Theory Dynam. Systems 22 (2002),
3, 757-782.
[16] Mikhail Gordin, Michel Weber. On the almost
sure central limit theorem for a class of Z d
-actions. J. Theoret. Probab. 15 (2002):
2,
477-501.
[17] Mikhail Gordin. Double extensions of dynamical
systems and the construction of mixing filtrations. II.
Quasihyperbolic automorphisms of tori Zap. Nauchn. Sem.
S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 260 (1999)
Veroyatn. i Stat. 3:103-118, 318-319 (Russian). Engl.
transl. in J. Math. Sci. (N. Y.) 109 (2002): 6,
2103-2114.
[18] Denker M., Gordin M. Gibbs measutres for fibred
systems. Advances in
Mathematics, 148, 1999, no.2, 161-192.
[19] Denker M., Gordin M. The central limit theorem
for random perturbations of rotations. Probab. Theory and
Related Fields. 111 1998, no. 1, 1-16.
[20] Gordin M. I. Double extensions of dynamical
systems and the construction of mixing filtrations. (Russian)
Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.
(POMI) 244 (1997), Veroyatn. i Stat. 2,
61-72, 330-331; transl. in J. Math. Sci. (New York) 99 (2000),
no. 2,
1053-1060.
[21] Denker, Manfred; Gordin, Mikhail. Remarks on
Gibbs measures for fibred systems. Problems on complex
dynamical systems (Kyoto, 1997). Su-rikaisekikenkyu-sho
Ko-kyu-roku No. 1042 (1998), 1-10.
[22] Gordin M. I. Extensions of dynamical systems
and the martingale approximation method. (Russian) Zap.
Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 216
(1994), Problemy Teorii Veroyatnost. Raspred. 13,
10-19, 161; translation in J. Math. Sci. (New York) 88 (1998),
no. 1, 7-12.
[23] Gordin M. I. Some remarks on homoclinic groups
of hyperbolic auto- morphisms of tori. (Russian) Zap.
Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 223
(1995), Teor. Predstav. Din. Sistemy, Kombin. i Algoritm.
Metody. I, 140-147, 339; translation in J. Math. Sci. (New York)
87 (1997), no. 6, 4067-4071.
[24] M.I. Gordin, B.A. Lifshits. Martingale approach
in the limit theorems for random walks. In book Limit
theorems for functionals of random walks. Proceedings of the
Steklov Mathematical Institute, 195 (1994), by
A.N. Borodin, I.A. Ibragimov, St Petersburg, Nauka. English
transl. (1995) AMS, Providence, Rhode Island USA, 164-176.
[25] M. Gordin. Homoclinic approach to the Central
Limit Theorem for dynamical systems. "Doeblin and Modern
Probability" (Blaubeuren,
1991), 149-162, Contemp. Math., 149, Amer. Math. Soc.,
Providence, RI, 1993.
[26] M.I. Gordin. A homoclinic version of the
Central Limit Theorem.
Zap. Nauchn. Sem. LOMI (Scientific Seminars Notes of the
Leningrad Branch of Steklov Mathematical Institute)184 (1990),
80-91 (Russian). Engl. transl. in Journ. of Math. Sci. 68:
(1994), no.4, 451-458.
Markov processes. Teorija verojatnostej i ejo prim. 23
(1978): 4 (1978),
147-148 (Russian).
Markov processes. Doklady Akademii Nauk SSSR, 239 (1978),4,
766-
767 (Russian). Engl. transl. in Soviet Math. Dokl., 19 (1978),
2,
392-394.
[29] M.I. Gordin. On the behavior of the variances of sums composed from stationary connected random variables Teorija verojatnostej i ejo prim.
its applications.
[30] M.I. Gordin. Exponentially fast mixing. Doklady
Akademii Nauk SSSR
[31] M.I. Gordin, M.Kh. Resnik. The Law of the Iterated Logarithm for the denominators of continued fractions. Vestnik Leningrad Univ. 25 (1970): 13, 28-33 (Russian). Engl. transl. in Vestnik Leningrad Univ. Math. 3 (1976), 207-213.
Doklady Akademii Nauk SSSR 188 (1969): 4, 739-741
(Russian) Engl. transl. in Soviet Math. Dokl. 10(1969):
5, 1174-1176.
[33] M.I. Gordin. Stochastic processes generated by
number - theoretic endomorphisms. Doklady Akademii Nauk
SSSR 182 (1968): 5, 1004-
1006 (Russian). Engl. transl. in Soviet Math. Dokl. 9 (1968):
5,
1234-1237.
1. M. Gordin. Stationary fields, martingale approximation,
tensor spaces and von Mises statistics. 10th Vinius
Conference on Probability Theory and Mathematical Statistics,
28th June - 2th July, 2010, Vilnius, Lithuania. Abstracts of
Communications. Vilnius, 2010, P. 40-41.
2. M. Gordin. An application of multiparameter martingale
approximation. Prague Stochastics 2010. Book of Abstracts.
Prague, P. 87.
3. Mikhail Gordin, Michel Weber. Degeneration in the central
limit theorem for a class of multidimensional actions.
Abstracts of 9-th Vilnius Conference on Probab. Theory, 2006, P.
151.
4. Mikhail Gordin.Homoclinic approach to the central limit
theorem. 11th Prague Conference on Information Theory,
Statistical Decision Functions and Random Processes, Abstracts,
Prague, 1990.
6. M.I. Gordin. Homoclinic transformation and the central
limit theorem.(Russian) Abstracts of 5-th Vilnius
Conference on Probab. Theory and Math. Stat., vol. 3,
Vilnius 1989, P.156-157.
7. Mikhail Gordon. Ergodic properties of a class of a
queueing systems. Abstracts of Communications of the
First World Congress of the Bernoulli Society, vol. 2.
Moscow, 1986, P. 546.
8. M.I. Gordin, B.A. Lifshits. Central Limit Theorem for
periodograms of stationary sequences related to a class of
Markov chains. (Russian). Abstracts of 4-th Vilnius
Conference on Probab. Theory and Math. Stat., vol. 1,
Vilnius 1985, P.182-183.
9. M.I. Gordin, B.A. Lifshits. A remark on Markov processes
with normal transition operators. The Third Vilnius
International Conference on Probability and Mathematical
Statistics, Absracts of Communications. vol. 1, Vilnius,
1981,147-148 (Russian).
10. M.I. Gordin. The Central Limit Theorem for stationary
processes without the finiteness of variance assumption. The
First Vilnius International Conference on Probability and
Statistics, Absracts of communications. vol. 1, Vilnius,
1973,173-174 (Russian).
11. Yu.A. Davydov, M.I. Gordin, I.A. Ibragimov, V.N. Solev. S
tationary processes: limit theorems, regularity conditions.
In "Soviet-Japanese Symposium on Probability Theory",
Novosibirsk, 1989.
1. (with Michel Weber) Degeneration in the Central Limit
Theorem for a class of multidimensional actions.
2. The Central Limit Theorem for stationary Markov chains
with normal transition operator.
3. (with Friedrich Götze) Wigner matrix ensembles via
external source matrix models.
4. Limit theorems for hyperbolic non-homomorphic actions on
tori.
There are more than 20 papers and technical reports (1970 - 1992) on statistical theory of measurements and mathematical models of parallelism and interaction.
In 2014 we celebrate the 70th birthday of Mikhail Iosifovich Gordin, a Senior Researcher at the Laboratory of Statistical Methods of POMI RAS, a faculty member of the Probability and Mathematical Statistics Department, Mathematics and Mechanics Faculty of the St. Petersburg State University, a permanent active participant of the St. Petersburg Seminar on Probability Theory and Mathematical Statistics.
Mikhail Iosifovich was born on September 9, 1944 in Leningrad. In 1966, he graduated from the Faculty of Mathematics and Mechanics of Leningrad University; from 1966 to 1969 was a graduate student at Leningrad University under the guidance of I.A. Ibragimov, and after graduation began to work at the Research Institute for Electrical Measurements (VNIIEP, now Electromera). Since 1992, M.I. has worked at the Laboratory of Statistical Methods of POMI of the Russian Academy of Sciences, and at the same time he teaches at the Department of Probability Theory and Mathematical Statistics of the Mathematics and Mechanics Faculty of St. Petersburg State University.
Mikhail Iosifovich is a person of wide interests and general high culture, but above all he remains a mathematician. It is impossible to give a detailed account of M.I.’s remarkable results in this short text, and we only briefly outline some of the main directions of his research.
1. M.I. developed methods to study the important and interesting classes of stationary random processes (measure-preserving transformations) satisfying strong exponentially fast mixing conditions. Such assumptions allow one to prove almost the same limit theorems for such processes as for independent random variables. Relevant examples were given in the first published work of M.I., in which he considered transformations $T$ now called the Rényi transformations, of the interval $[0,1]$ to itself defined by the equation $Tx = \{f^{-1}x\}$ where $f(x)$ is a monotone positive function on $[0,1]$. Rényi showed that under certain assumptions on $f$, there exists a measure absolutely continuous with respect to the Lebesgue measure such that $T$ is an ergodic endomorphism for this measure. M.I. proved that, under some additional assumptions on $f$, the stationary processes $X_n =f (T^n x)$ satisfy very strong exponentially fast mixing conditions. When $f(x) = 10x$, the powers of $T$ generate the decimal expansion of $x$, while with $f=1/x$ they generate the continued fraction expansion of $x$. Note that these results of M.I. for $f(x) = 1/x$ imply the celebrated results of R.O. Kuzmin and P. Lévy on continued fractions.
Later, M.I. obtained suﬃciently general results that guarantee similar fast mixing for measure space endomorphisms and the stationary processes generated by them. As examples, M.I considered transformations of the form $\{\theta x\}$ and Jacobi–Perron transformations.
2. A significant part of M.I.’s work was research on limit theorems for dependent random variables. M.I. created a very powerful method, which allows one to prove limit theorems for suﬃciently general stationary processes. He was able to identify the role of coboundaries and martingale diﬀerences as canonical representatives of the corresponding cohomology classes. This led to the development of the martingale approximation method, which was first introduced by M.I. as a general method in a short publication in 1969. Since then the
technique has been widely developed and used by M.I. and other mathematicians, and is now considered to be one of the main methods for proving limit theorems. Here are two quotes that show how this method is perceived by mordern researchers. They are taken from two articles published in 2014, in one of, as they say nowadays, prestigious journals; one of these articles was written by a very prominent mathematician.
“Since the seminal paper of Gordin in 1969, approximation via a martingale is known to be a nice method to derive limit theorems for stochastic processes.”
“Our methods rely on a martingale approximations approach which has played a decisive role in most proofs of the central limit theorem during the last 50 years.”
We have already mentioned that M.I. successfully developed his own method and applied it to the study of asymptotic problems of probability theory and the theory of dynamical sys- tems. One of his latest very interesting results in this direction (joint with Manfred Denker), published in 2014 in the journal Probability Theory and Related Fields, contains deep study of the asymptotic behavior of von Mises statistics for measure-preserving transformations. Namely, this paper deals with the limiting behavior of the forms $\sum_{0\le i_1 \cdots \le i_d \le n}f (T^{i_1} x, \dots,T^{i_d}x)$ where $T$ is a measure-preserving transformation. This paper contains many new ideas and probably will have a significant influence on the study of similar problems.
Finally we mention only briefly several other achievements of M.I.
3. The definition of homoclinic transformations and study of their role in the limit theo- rems for hyperbolic dynamical systems.
4. The development of extension methods for some classes of partially hyperbolic systems.
A characteristic feature of M.I.’s work is a great sense of the unity of mathematics: his work, at first sight unexpectedly, but very naturally, uses methods and concepts from areas of mathematics that would seem quite distant from the main problem. All participants of the St. Petersburg seminar also know that, no matter what the topic of the next talk would be (as long as it is interesting), M.I. would come up with deep questions and insightful comments.
The editorial board and authors of this volume congratulate M.I. on his birthday, wish him good health, many years of active creative life, and dedicate this volume of the Notes of the Seminars of POMI to him.