This talk is about the variance of partial sums and the central limit theorem for additive functionals of stationary Markov chains with general state space, by using a new idea involving conditioning with respect to both the past and future of the chain. This idea was recently used to show that any stationary and ergodic Markov chain with $var(S_{n})/n$ uniformly bounded, satisfies a $\sqrt n$-central limit theorem with a random centering. We shall discuss now new projective conditions for the linearity in $n$ of $var(S_{n})$ which lead to a CLT without random centering. Several open problems will be pointed out.

For many families of $x\times n$ random hermitian matrices $X_n$ with eigenvalues $\lambda_1,\dots, \lambda_n$, their random spectral measures $\mu_n(\cdot) =\sum_{j=1}^n 1_{\lambda_j\in (\cdot)}$ become close to (possible non-convergent) sequences of deterministic measures $\nu_n$, as the dimension $n$ increases. I will discuss the "universality" and the "equations" that should be included in the concept of "deterministic equivalence". This talk was inspired by discussions with Jack W. Silverstein.

Local central limit theorem is much more demanding than the usual CLT. We shall present a local limit theorem for linear fields of random variables constructed from independent and identically distributed innovations, each with finite second moment. When the coefficients are absolutely summable we do not restrict the region of summation. However, when the coefficients are only square-summable we add the variables on unions of rectangle and we impose a regularity condition on the coefficients, depending on the number of rectangles considered. Our results are new also for the dimension 1, i.e. for linear sequences of random variables. The examples include the fractionally integrated processes.

Joint work with Timothy Fortune and Hailin Sang.

I will review recent advances on the so-called Karlin random fields. The talk will highlight on a) a correlated random walk that serves as a discrete counterpart to the fractional Brownian motion with Hurst index $<1/2$, and b) a natural extension to manifold-indexed ones that extends the well-known Lévy Brownian/stable fields (these are random fields indexed by $\mathbb R^d$, $\mathbb S^d$ and $\mathbb H^d$).

The talk is based on a series of joint works with Oliver Durieu, Zuopeng Fu, and Gennady Samorodnitsky.

In this talk, fast and exact simulation methods for fractional Brownian motions and fractional Levy Brownian fields will be reviewed. They are needed in our recent investigations for simulations for Karlin stable set-indexed processes.

The speaker will discuss various standard numerical algorithms acting on random data. It turns out that the fluctuations in the time to compute the solution of the problem at hand to a desired accuracy, turn out to be universal, independent of the statistical assumptions on the data. Some of the results discussed are numerical and experimental, and some are proved rigorously. This is joint work with various authors, C. Pfrang, G. Menon, S. Olver, S. Miller, and (mostly) with Tom Trogdon.

Nonintersecting Brownian bridges on the unit circle form a determinantal stochastic process exhibiting random matrix statistics for large numbers of walkers. We investigate the effect of adding a drift term to walkers on the circle conditioned to start and end at the same position. We use Riemann-Hilbert analysis of a family of discrete orthogonal polynomials with a complex weight to compute the asymptotic distribution of total winding numbers in the scaling regime in which the expected total winding is finite. Furthermore, we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process expressed in terms of generalized Hastings-McLeod functions. We further investigate the large-degree asymptotic behavior of the generalized Hastings-McLeod functions, which have arisen in a variety of integrable probability contexts. This is joint work with Karl Liechty (DePaul University) and Kurt Schmidt (University of Cincinnati).