Konstantin Tikhomirov

On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence

Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko–Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refinement of the rank one update approach used by Srivastava and Vershynin to produce non-asymptotic quantitative estimates. In other words we provide a new proof of the Bai and Yin theorem using basic tools from probability theory and linear algebra, together with a new extension of this theorem to random matrices with dependent entries.

On the norm of a random jointly exchangeable matrix

In this note, we show that the norm of an

On the Distance of Polytopes with Few Vertices to the Euclidean Ball

n\times n

On the spectral radius of a random matrix: an upper bound without fourth moment

n×n random jointly exchangeable matrix with zero diagonal can be estimated in terms of the norm of its

The smallest singular value of random rectangular matrices with no moment assumptions on entries

\lfloor n/2\rfloor \times \lfloor n/2\rfloor

Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

⌊n/2⌋×⌊n/2⌋ submatrix located in the top right corner. As a consequence, we prove a relation between the second largest singular values of a random matrix with constant row and column sums and its top right