Let K be an isotropic convex body in R n . Given ε > 0, how many independent points X i uniformly distributed on K are needed for the empirical covariance matrix to approximate the identity up to ε with overwhelming probability? Our paper answers this question from [12]. More precisely, let X ∈ R n be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector X is a random point in an isotropic convex body. We show that for any ε > 0, there exists C(ε) > 0, such that if N ∼ C(ε) n and (X i ) i≤N are i.i.d. copies of, with probability larger than 1 − exp(−c √ n).

We present a tail inequality for suprema of empirical processes generated by variables with finite ψ α norms and apply it to some geometrically ergodic Markov chains to derive similar estimates for empirical processes of such chains, generated by bounded functions. We also obtain a bounded difference inequality for symmetric statistics of such Markov chains.AMS 2000 Subject Classification: Primary 60E15, Secondary 60J05. * Research partially supported by MEiN Grant 1 PO3A 012 29. n i=1 Ef (X i ) 2 . Then for all η > 0 and t ≥ 0,where K 1 = 4, K 2 (η) = 2.5 + 32/η, K 3 = 5.4, K 4 (η) = 2.5 + 43.2/η. Similar, more refined results were obtained subsequently by Bousquet [2] and Klein and Rio [7]. The latter article contains an inequality for suprema of empirical processes with the best known constants.

Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for not necessarily Lipschitz functions f : R n → R with bounded derivatives of higher orders, which holds when the underlying measure satisfies a family of Sobolev type inequalitiesSuch Sobolev type inequalities hold, e.g., if the underlying measure satisfies the logSobolev inequality (in which case C( p) ≤ C √ p) or the Poincaré inequality (then C( p) ≤ C p). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of f . When the underlying measure is Gaussian and f is a polynomial (not necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erdős-Rényi random graphs, obtaining new estimates, optimal in a certain range of parameters.