Probability Theory Group, Institute of Mathematics, University of Warsaw

Scientific interests

High Dimensional Probability
Concentration of measure
Asymptotic theory of $U$-statistics
Random matrices
Markov chains


Institute of Mathematics
University of Warsaw
ul. Banacha 2
02-097 Warszawa
r "dot" surname "at" mimuw "dot" edu "dot" pl (work)
radek "dot" "surname" "at" o2 "dot" pl (private)

Please take a minute to follow the link in the picture to the right and visit the site of the Polish Humanitarian Organization. By clicking on the puppet's belly you help the children in need.

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Zajęcia 2016/2017

Konsultacje - piątki 14:15-15:45, p. 1220

Rachunek prawdopodobieństwa - proseminarium


Rachunek prawdopodobieństwa - bioinformatyka

Materiały dydaktyczne

Elementy teorii macierzy losowych - notatki

Rachunek prawdopodobieństwa (WNE) - notatki napisane z Adamem Osękowskim

Procesy stochastyczne II - notatki napisane z Witoldem Bednorzem

Równania różniczkowe zwyczajne - notatki prof. Feliksa Przytyckiego

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Scientific interests - a brief description

My research area is probability theory, with focus on high dimensional phenomena. I am particularly interested in concentration inequalities and their applications to limit theorems. Other fields I am interested in are Random Matrix Theory, Markov chains, the theory of $U$-statistics and branching particle systems.


Research articles

The main links below lead to journal versions of the articles. The arXiv versions may differ (in fact do differ) from the final ones. You may also visit my MathSciNet or Google Scholar profiles.

1. Logarithmic Sobolev Inequalities and Concentration of Measure for Convex Functions and Polynomial Chaoses, Bulletin Polish Acad. Sci. Math. 53 (2005), 221-238. arXiv version

2. Moment Inequalities for U-statistics, Ann. Probab. 2006, Vol. 34, No. 6, 2288-2314. arXiv version

3. With R. Latała, The LIL for canonical U-statistics, Ann. Probab. 2008, Vol. 36, No. 3, 1023-1058. arXiv version

4. With R. Latała, The LIL for U-statistics in Hilbert spaces, J. Theoret. Probab. 21 (2008), 704-744. arXiv version

5. A tail inequality for suprema of unbounded empirical processes with applications to Markov chains, Electronic Journal of Probability, Vol. 13 (2008), 1000-1034.

6. A few remarks on the operator norm of random Toeplitz matrices, J. Theoret. Probab, Volume 23 (2010), Number 1, 85-108. arXiv version

7. With O. Guédon, A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Smallest singular value of random matrices with independent columns, Comptes Rendus Mathematique Vol. 346, Issues 15-16, 2008, 853-856

8. With A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles. J. Amer. Math. Soc. 23 (2010), 535-561. arXiv version

9. With A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling. Constructive Approximation Vol. 34, No. 1, 61-88. arXiv version

10. With R. Latała, Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails. Annales de l’Institut Henri Poincaré. Probab. Statist. Volume 48, Number 4 (2012), 1103-1136. arXiv version

11. With O. Guédon, A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Condition number of a square matrix with i.i.d. columns drawn from a convex body. Proc. Amer. Math. Soc. 140 (2012), 987-998. pdf file

12. With A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Sharp bounds on the rate of convergence of the empirical covariance matrix. Comptes Rendus Mathematique, Volume 349, Issues 3-4, February 2011, Pages 195-200. arXiv version

13. On the Marchenko-Pastur and circular laws for some classes of random matrices with dependent entries Electronic Journal of Probability, Vol. 16 (2011) 1065-1095.

14. With R. Latała, A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Geometry of log-concave Ensembles of random matrices and approximate reconstruction. Comptes Rendus Mathematique, Volume 349, Issues 13–14, July 2011, Pages 783–-786. This is an announcement (without proofs) of the results contained in the following two articles. arXiv version

15. With R. Latała, A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Chevet type inequality and norms of submatrices. Studia Math. 210 (2012), 35-56. arXiv version

16. With R. Latała, A. Litvak, A. Pajor and N. Tomczak-Jaegermann, Tail estimates for norms of sums of log-concave random vectors (To appear in Proceedings of the London Mathematical Society). arXiv version

17. With P. Miłoś, CLT for Ornstein-Uhlenbeck branching particle system. Electronic Journal of Probability, Vol 20, Article 42 (2015). arXiv version

18. With P. Miłoś, U-statistics of Ornstein-Uhlenbeck branching particle system. Journal of Theoretical Probability. Volume 27, Issue 4, 1071-1111 (2014).

19. With W. Bednorz, Exponential Concentration Inequalities for Additive Functionals of Markov Chains. ESAIM: PS, 19, 440-481 (2015). arXiv version

20. With R. Latała, A. Litvak, K. Oleszkiewicz, A. Pajor and N. Tomczak-Jaegermann, A Short Proof of Paouris' Inequality Canadian Mathematical Bulletin 57, 3-8 (2014). arXiv version

21. With O. Guédon, R. Latała, A. Litvak, K. Oleszkiewicz, A. Pajor and N. Tomczak-Jaegermann, Moment estimates for convex measures. Electronic Journal of Probability, Vol. 17, Article 1

22. The operator norm of random rectangular Toeplitz matrices. High Dimensional Probability VI, The Banff Volume. Progress in Probability, Vol. 66, 247-260 (2013). .pdf file

23. Some remarks on the Dozier-Silverstein Theorem for random matrices with dependent entries. Random Matrices: Theory and Applications, Volume 02, Issue 02, April 2013

24. With D. Chafaï, Circular law for random matrices with unconditional log-concave distribution. Communications in Contemporary Mathematics Volume 17, Issue 04 (2015). arXiv version

25. With P. Wolff, Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order. Probability Theory and Related Fields. Vol. 162, Issue 3, 531-586 (2015). arXiv version

26. With W. Bednorz, Some remarks on MCMC estimation of spectra of integral operators. Bernoulli, Vol. 21, No. 4, 2073-2092 (2015). arXiv version

27. With D. Chafaï and P. Wolff, Circular law for random matrices with exchangeable entries. Random Structures and algorithms, Volume 48, Issue 3 (2016). arXiv version

28. With W. Bednorz, Orlicz integrability of additive functionals of Harris ergodic Markov chains. High Dimensional Probability VII, Cargèse Volume, Progress in Probability, vol 71. Birkhäuser, Cham, 295-326 (2016). arXiv version

29. A note on the Hanson-Wright inequality for random vectors with dependencies. Electron. Commun. Probab. Volume 20 (2015), paper no. 72. arXiv version

30. With R. Latała, Z. Puchała, K. Życzkowski, Asymptotic entropic uncertainty relations. J. Math. Phys. 57, 032204 (2016). arXiv version

31. With M. Strzelecki, Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions. Studia Mathematica 230 (2015), 59-93. arXiv version

32. With W. Bednorz and P. Wolff, Moment estimates implied by modified log-Sobolev inequalities. To appear in ESAIM: PS. arXiv version

33. A note on the sample complexity of the Er-SpUD algorithm by Spielman, Wang and Wright for exact recovery of sparsely used dictionaries. Journal of Machine Learning Research, 17(177):1-18 (2016). arXiv version

34. Metric and classical fidelity uncertainty relations for random unitary matrices. Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 10 (2017). arXiv version

35. On the convex Poincaré inequality and weak transportation inequalities (preprint). arXiv version

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Short CV

Academic degrees

2016 Habilitation, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw. Random matrices with dependent coefficients

2007 PhD in Mathematics, Institute of Mathematics, Polish Academy of Sciences. Thesis: U-statistics. Inequalities and laws of the iterated logarithm. Advisor: Prof. Rafał Latała.

2003 MSc in Mathematics, University of Warsaw and Vrije Universiteit van Amsterdam (within the Joint Master's Programme). Master's thesis: The entropy method and concentration of measure in product spaces (at the University of Warsaw: "Koncentracja miary") written under supervision of prof. Rafał Latała (Poland) and prof. Aad van der Vaart (the Netherlands).

2001 Bachelor's thesis at the University of Warsaw: "Symetryzacje dwupunktowe i radialne, a nierówność izoperymetryczna" (Two point and radial symmetrizations and the isoperimetric inequality) written under supervision of prof. Rafał Latała


A member of the Polish Mathematical Society


2012 Scholarship for Outstanding Young Scientists, Ministry of Science and Higher Education

2010 Nomination for the International Banach Prize

2010 Wojtek Pulikowski Lectureship at the Adam Mickiewicz University in Poznań

2010 Foundation for Polish Science Scholarship (special distinction)

2009 Kazimierz Kuratowski Award

2009 Foundation for Polish Science Scholarship

2008 Polish Mathematical Society Award for Young Mathematicians

2007 Marek Wacławek Foundation Award

2007 Stanisław Saks Scholarship given by the Foundation of Mathematicians of the City of Wrocław

1996-1998 Polish Children Found Scholarship

Foreign Stays

2010 A four month postdoctoral stay at the Fields Institute in Toronto

2007-2008 A one year postdoctoral stay at the Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada (Pacific Institute for the Mathematical Sciences Postdoctoral Fellowship)

2006 A three-month stay at the Institut Henri Poincaré in Paris during the special trimester "Phenomena in High Dimensions" (Marie Curie Fellowship)

2003 An eight-month stay at Vrije Universiteit van Amsterdam, Faculty of Science, Department of Mathematics within the Joint Master's Programme of VUA and University of Warsaw

2001 A four-month stay at Vrije Universiteit van Amsterdam, Faculty of Science, Department of Mathematics within the Socrates Erasmus Exchange Programme.


2012-2015 A member of the Kuratowski Award Jury (vice chairman in 2014, chairman in 2015)

2013 A member of the panel of experts in the Fuga grant competition (National Science Center, Poland)

2014 A member of the panel of experts in the Etiuda grant competition (National Science Center, Poland)


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Some links


Books and lecture notes

Probability in Banach Spaces by M. Ledoux and M. Talagrand at M. Talagrand's home page

Markov Chains and Stochastic Stability by S.P. Meyn and R. L. Tweedie

An Introduction to Random Matrices by G. Anderson, A. Guionnet and O. Zeitouni at O. Zeitouni's home page

Expander Graphs and their Applications by S. Hoory, N. Linial and A. Wigderson at A. Wigderson's web page

Martingales in Banach Spaces by G. Pisier

Riemann-Hilbert Problems and Multiple Orthogonal Polynomials by Arno Kuijlaars


Math blogs and popular science

What's new by Terence Tao

Combinatorics and more by Gil Kalai

Timothy Gowers' Weblog

Libres pensées... by Djalil Chafaï

Almost sure by George Lowther

Quanta Magazine


My old web page

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