Radosław Adamczak

Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles

Let K be an isotropic convex body in Rn. Given e > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to e with overwhelming probability? Our paper answers this question from [12]. More precisely, let X ∈ Rn 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 e > 0, there existsC(e) > 0, suﰁch that if N ∼ C(e)n and (Xi)i≤N are i.i.d. copies of ﱞﱞ1 N ﱞﱞ X, then ﱞN i=1 Xi ⊗ Xi − Idﱞ ≤ e, with probability larger than 1 − exp(−c√n).

Moment inequalities for U-statistics

We present moment inequalities for completely degenerate Banach space valued (generalized) U-statistics of arbitrary order. The estimates involve suprema of empirical processes which, in the real-valued case, can be replaced by simpler norms of the kernel matrix (i.e., norms of some multilinear operators associated with the kernel matrix). As a corollary, we derive tail inequalities for U-statistics with bounded kernels and for some multiple stochastic integrals.

Circular law for random matrices with exchangeable entries

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non-Hermitian counterpart of a result of Chatterjee on the semi-circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on Hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest.

A Short Proof of Paouris' Inequality

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm

Tail estimates for norms of sums of log-concave random vectors

|X|

U-Statistics of Ornstein–Uhlenbeck Branching Particle System

of an isotropic log-concave random vector

Circular law for random matrices with unconditional log-concave distribution

X\in\R^n

Condition number of a square matrix with i.i.d. columns drawn from a convex body

, stating that for every

Geometry of log-concave Ensembles of random matrices and approximate reconstruction

t\geq 1

Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails

,