This talk is based on a joint paper with Jack W. Silverstein. We study largest singular values of large random matrices, each with mean of a fixed rank $K$. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest $K$ singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of the entries of the matrix, and a term negligible in probability. The representation can be used to establish asymptotic normality of the largest singular values for random matrices with means that have a block structure.

We shall review the notion of random sup-measures and their role in extreme value theory. A few families of random sup-measures that appeared in recently developed limit theorems are introduced.

We shall review the results related to martingale approximation for random sequences, then we will talk about the ortho-martingales approximation for random fields and introduce the corresponding results under projective criteria.

We will present a probabilistic proof of the Wong-Zakai theorem for stochastic heat equation by Hairer-Pardoux.

This is an expository talk on spectral density and its role in the investigations of variance of partial sums, Markov chain operators, Fourier transforms and random matrices.

I will discuss some facts about the substitutes for the matrix ansatz for ASEP in the "singular case", where the standard form of the matrix ansatz of Derrida et al. (1993) does not apply. This is joint work in progress with Marcin Swieca.

Typical results on the eigenvalues of large dimensional random matrices are limit theorems, as the dimension $n$ of the matrix increases, on the empirical distribution of the eigenvalues to a nonrandom limit. Assumptions are made on nonrandom quantities used in constructing the matrix. There are though examples where the eigenvalues of the random matrix cannot possibly converge to one nonrandom quantity. However there are results where, although there is no limiting eigenvalue distribution, for each $n$ the random empirical eigenvalue distribution is close to a nonrandom one which is a function of the nonrandom quantities. This nonrandom distribution has been called a deterministic equivalent to the random one.

This talk will present two examples of ensembles with no general nonrandom limiting eigenvalue distributions, but possess deterministic equivalents, and will present joint work with Wlodek Bryc on the ensemble $(1/N)(D_n\circ X_n)(D_n\circ X_n)^*$, where $D_n$ and $X_n$ are $n\times N$, $D_n$ consists of nonnegative entries, $X_n$ consists of independent standardized random variables, and $\circ$ denotes the Hadamard (entrywise) product.

Two integrable random vectors $\xi$ and $\eta$ in the Euclidean space are said to be zonoid equivalent if their projections on each given direction share the same first absolute moments. This is equivalent to the equality of expectations of symmetric random segments $[-\xi,\xi]$ and $[-\eta,\eta]$ determined by these vectors. This talk discusses a generalisation of this concept based on rescaling a convex body $K$ by applying the diagonal matrix with the entries given by a random vector. The particular attention is devoted to analysing the cases when the equality of expectations of $\xi K$ and $\eta K$ implies the zonoid equivalence of $\xi$ and $\eta$. Applications to geometric interpretation of one-sided stable laws are also discussed.

Joint work with Felix Nagel (Bern).

My talk will be a continuation of Wlodek's talk. I will sketch a proof of theorem that connects ASEP with Askey-Wilson polynomials. This is joint work with Wlodek Bryc.

Social network modeling provides plenty of data but realistic models for network growth must be simple if mathematical results are expected. We have used preferential attachment (PA) models with a small number of parameters in an attempt to strike a balance between the mathematics and the statistical fitting. The PA models struggle to match the data but provide a context in which to test methods and analyze estimation techniques. Numerical summaries of network characteristics are often estimated using methods imported from classical statistics without real justification. For example, the Hill estimator coupled with a minimum distance threshold selection technique are commonly used. We discuss some attempts to justify and understand these estimation methods in the context of PA models. Without a model and its properties, there is no way to understand the limitation of estimation methods.

A random lattice is a random element of $SL(n,\mathbb Z) \ SL(n,\mathbb R)$ with respect to the natural probability measure induced by the Haar measure on $SL(n,\mathbb R)$. For various reasons one is interested in studying the statistics regarding the lengths of its vectors e.g. the distribution of the length of the shortest nonzero vector (related to sphere packing), the distribution of the gap between the lengths of $n$-th and ($n+1$)-st vectors (i.e. spectral gap of an $n$-dimensional torus). I will give an introduction to the methods, results, and subtleties in this area.

Many statistical analyses are characterized by how often a procedure works: how often an interval covers a true value, a null hypothesis is rejected, an item is correctly classified, etc. But assessing how often a procedure works differs from assessing the evidence in a data set. Understanding the difference is prerequisite to understanding what matters in a given analysis: the procedure, the evidence, or both.

We begin with several examples that illustrate the difference between assessing evidence and assessing procedures. Then we work carefully through another example that explains the fundamental difference: assessing procedures requires and assessing evidence forbids averaging over other data sets. Along the way we introduce foundational principles of evidence and inference, namely the Conditionality Principle (CP) and the Sufficiency Principle (SP), and show how CP and SP together imply the Likelihood Principle.