Georgia Institute of Technology College of Sciences

In this talk we construct a net around the unit sphere with strong properties. We show that with exponentially high probability, the value of |Ax| on the sphere can be approximated well using this net, where A is a random matrix with independent columns. We apply it to study the smallest singular value of random matrices under very mild assumptions, and obtain sharp small ball behavior. As a partial case, we estimate (essentially optimally) the smallest singular value for matrices of arbitrary aspect ratio with i.i.d. mean zero variance one entries. Further, in the square case we show an estimate that holds only under simply the assumptions of independent entries with bounded concentration functions, and with appropriately bounded expected Hilbert-Schmidt norm. A key aspect of our results is the absence of structural requirements such as mean zero and equal variance of the entries.

We discuss a problem of asymptotically efficient (that is, asymptotically normal with minimax optimal limit variance) estimation of functionals of the form $\langle f(\Sigma), B\rangle$ of unknown covariance $\Sigma$ based on i.i.d.mean zero Gaussian observations $X_1,\dots, X_n\in {\mathbb R}^d$ with covariance $$\Sigma$. Under the assumptions that the dimension $d\leq n^{\alpha}$ for some $\alpha\in (0,1)$ and $f:{\mathbb R}\mapsto {\mathbb R}$ is of smoothness $s>\frac{1}{1-\alpha},$ we show how to construct an asymptotically efficient estimator of such functionals (the smoothness threshold $\frac{1}{1-\alpha}$ is known to be optimal for a simpler problem of estimation of smooth functionals of unknown mean of normal distribution).

The proof of this result relies on a variety of probabilistic and analytic tools including Gaussian concentration, bounds on the remainders of Taylor expansions of operator functions and bounds on finite differences of smooth functions along certain Markov chains in the spaces of positively semi-definite matrices.

Stein's method is a powerful technique to quantify proximity between probability measures, which has been mainly developed in the Gaussian and the Poisson settings. It is based on a covariance representation which completely characterizes the target probability measure. In this talk, I will present some recent unifying results regarding Stein's method for infinitely divisible laws with finite first moment. In particular, I will present new quantitative results regarding Compound Poisson approximation of infinitely divisible laws, approximation of self-decomposable distributions by sums of independent summands and stability results for self-decomposable laws which satisfy a second moment assumption together with an appropriate Poincaré inequality. This is based on joint works with Christian Houdré.

(Based on joint work with Cécile Mailler)Consider a stochastic process that behaves as a d-dimensional simple and symmetric random walk, except that, with a certain fixed probability, at each step, it chooses instead to jump to a given site with probability proportional to the time it has already spent there. This process has been analyzed in the physics literature under the name "random walk with preferential relocations", where it is argued that the position of the walker after n steps, scaled by log(n), converges to a Gaussian random variable; because of the log spatial scaling, the process is said to undergo a "slow diffusion". We generalize this model by allowing the underlying random walk to be any Markov process and the random run-lengths (time between two relocations) to be i.i.d.-distributed. We also allow the memory of the walker to fade with time, meaning that when a relocations occurs, the walker is more likely to go back to a place it has visited more recently. We prove rigorously the central limit theorem described above by associating to the process a growing family of vertex-weighted random recursive trees and a Markov chain indexed by this tree. The spatial scaling of our relocated random walk is related to the height of a typical vertex in the random tree. This typical height can range from doubly-logarithmic to logarithmic or even a power of the number of nodes of the tree, depending on the form of the memory.