### Probability Seminar at University of CincinnatiSpring 2016

This semester the regular meeting time of our seminar is Tues 2:30-3:25pm, Seminar Room, French Hall 4206. Please email yizao.wang@uc.edu if you want to be added to our email list or have any questions.
• Jan 22, Fri, 2:15-3:15pm, 716 Swift Hall. Colloquium. Joseph Najnudel, Université Paul Sabatier (Toulouse III), France.

A random holomorphic function related to Haar-distributed random unitary matrices

We show that after suitable rescaling, the characteristic polynomial of a Haar-distributed unitary matrix converges in law to a random holomorphic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. The scaling corresponds to the microscopic level, which means that we consider the characteristic polynomial at points at distances of order 1/n to each other. Some properties of our random holomorphic function are investigated: in particular, we get a determinantal formula for the moments of its ratios at different points. We also conjecture that our function is related to the behavior of the Riemann zeta function near the critical line.

• Jan 27, Wed, 4-5pm, 620 Swift Hall. Colloquium. Yu-Ting Chen, Harvard University.

Stochastic interaction systems on graphs

For more realistic modeling, there have been significant interests for the use of general graphs for interacting particle systems arising from biological or social contexts. However, the generality of spatial structure can lead to fundamental issues. They include the missing link to the stochastic PDE method that can give very detailed and clean information of interacting particle systems after rescaling, and the question of which graph parameters are only essential to describe certain probability laws of interest.

In this talk, I will discuss voter models and related methods, with an emphasis on the context of general graphs. I will demonstrate recent progress for some questions from Aldous for voter models and the so-called benefit-to-cost ratios first discovered by Ohtsuki, Hauert, Lieberman, and Nowak for evolutionary games that are variants of voter models. In particular, I will explain some diffusion approximation results for voter models on general graphs, which may provide new insights for related interacting particle systems on large graphs.

• Jan 29, Fri, 3:40-4:40pm, 800 Swift Hall, Colloquium. Bartłomiej Siudeja, University of Oregon.

Geometric measurements vs. heat and sound

The interface between stochastic processes and partial differential equations is an exciting area of mathematics employing techniques from many other fields. A common thread is the goal of relating spectral and other properties of the objects of study to properties of the domain on which they live. In particular, how do the size and shape of the domain influence the eigenvalues and eigenfunctions of the Laplace operator with a prescribed boundary condition? Can one bound spectral functionals using area, perimeter, diameter and other intrinsically geometric properties of the domain? Finally, how can one generalize the theory from Brownian motion (the Laplacian) to a larger class of stochastic processes? The talk will be accessible to a broad audience. All concepts will be related to physical phenomena, including heat retention and vibrational frequencies associated with a domain.

• Feb 2. No seminar

• Feb 9. canceled

• Feb 16. Wlodek Bryc.

Large deviations for the number of blocks of random non-crossing partitions

Additive statistics on noncrossing paritions are studied by Andu Nica (U of Waterloo) and his collaborators. I will explain the setup and present a result on large deviations for one such statistics.

• Feb 23. Yizao Wang.

The local structure of $q$-Gaussian processes

The local structure of $q$-Ornstein-Uhlenbeck processes and $q$-Brownian motions are investigated, for all $q\in(−1,1)$. These are the classical Markov processes corresponding to the noncommutative $q$-Gaussian processes. These processes have discontinuous sample paths, and the local small jumps are characterized by tangent processes. It is shown that for all $q\in(−1,1)$, the tangent processes at inner domain are scaled Cauchy processes possibly with drifts, and the tangent processes at the boundary of the domain are related to the free $1/2$-stable law via Biane's construction.

Joint work with Wlodek Bryc.

Random fields, spectral density and empirical spectral distribution

In this talk, we will survey some recent results on the empirical eigenvalue distribution of symmetric matrices with dependent entries, selected from regular random fields. It will be pointed out that, in many situations of interest, the limiting spectral measure always exists and depends only on the spectral density of the random field. The strength of the dependence is not important; the field can have long or short range memory and no rate of convergence to zero of the covariances is imposed. We characterize this limit in terms of the Stieltjes transform, via a certain equation involving the spectral density of the field. If the entries of the matrix are square integrable functions of an independent field, the results hold without any other additional assumptions.

• Mar 8. No event.

• Mar 15. 2-3pm, 220 WCharlton. Dissertation Defense. David Barrera.

Quenched Asymptotics for the Discrete Fourier Transforms of a Stationary Process

• Mar 17, Thurs 4-5pm, 140 WCharlton. Colloquium. Elizabeth Meckes, Case Western Reserve University.

Projections of probability distributions: A measure-theoretic Dvoretzky theorem

Dvoretzky's theorem tells us that if we put an arbitrary norm on $n$-dimensional Euclidean space, no matter what that normed space is like, if we pass to subspaces of dimension about $\log(n)$, the space looks pretty much Euclidean. A related measure-theoretic phenomenon has long been observed: the (one-dimensional) marginals of many natural high-dimensional probability distributions look about Gaussian. A natural question is whether this phenomenon persists for $k$-dimensional marginals for $k$ growing with $n$, and if so, for how large a $k$? In this talk I will discuss a result showing that the phenomenon does indeed persist if $k$ less than $2\log(n)/\log(\log(n))$, and that this bound is sharp (even the $2!$). The talk will not assume much background beyond basic probability and analysis; in particular, no prior knowledge of Dvoretzky's theorem is needed.

• Mar 18, Fri 10-11am, Lindner College of Business, Room 301. Arup Bose, ISI, India.

High-dimensional linear time series

How does one study the sample autocovariance matrices of linear time series when the dimension and the sample size both become large? One way to do this is to study the spectral distribution of these matrices or their polynomials. We develop the large sample behavior of these spectral distributions. The limit distributions often cannot be computed explicitly. Nevertheless, these results can be used in statistical inference problems such as estimation of high dimensional parameter matrices, determination of the order of the process and testing if the process is a white noise. This is a relatively new area of research and has plenty of potential applications in the study of high dimensional time series.

• Mar 22. UC spring break. No seminar.

• Mar 29. Florence Merlevède, Université Marne-la-Vallée, France.

Deviation and moment inequalities for dependent sequences and applications to intermittent maps

In this talk, we give a new deviation inequality for the maximum of partial sums of functions of alpha-dependent sequences. As a consequence, we extend the Rosenthal inequality of Rio (2000) for alpha-mixing sequences in the sense of Rosenblatt to the larger class of alpha-dependent sequences. Starting from the deviation inequality, we shall also exhibit sharp upper bounds for large deviations. We illustrate our results through the example of intermittent maps of the interval, which are not alpha-mixing in the sense of Rosenblatt.

• Apr 5. Jacek Wesołowski, Politechnika Warszawska, Poland.

Another version of the Matsumoto-Yor property on trees

Matsumoto-Yor property (and related characterization of the gamma and GIG laws) amounts for independence of $X+Y$ and $1/X-1/(X+Y)$ for independent $X$ and $Y$. Its symmetrized version means independence of $K_1$ and $K_2-1/K_1$ and independence of $K_2$ and $K_1-1/K_2$. In 2004, together with H. Massam we extended this symmetrized version (of property and characterization) to random vectors with structures "governed" by directed trees. In the talk we will present a parallel "tree goverened" approach to a property of independent Kummer and gamma variables: independence of $Y/(1+X)$ and $X(1+Y/(1+X))$ - discovered in 2015 by Pierre Vallois and his PhD student Marwa Hamza. Also the relative characterization will be presented. We do not understand what is the deep reason (if there is any) that such a parallel is possible. This is a joint work with Agnieszka Piliszek, a PhD student of mine.

• Apr 12. Rick Bradley, Indiana University.

Some mixing properties of some INAR models

In time series analysis of (nonnegative integer-valued) "count data", one sometimes uses "integer-valued autoregressive" (INAR) models, a variant of the original autoregressive models of classical time series analysis. The INAR models of order 1 with "Poisson innovations" satisfy (with exponential mixing rate) the $\rho^*$-mixing condition (the stronger variant of the usual $\rho$-mixing condition in which the two index sets are allowed to be "interlaced" instead of being restricted to "past" and "future"). That was shown in R.C. Bradley, Zapiski Nauchnyh Seminarov POMI 441 (2015) 56-72 (in the issue of that journal that was dedicated to the memory of Mikhail Gordin), and will be explained in this talk. Earlier, S. Schweer and C.H. Weiss, Comput. Statist. Data Anal. 77 (2014) 267-284, had already shown that those models (as well as some other closely related ones) satisfy absolute regularity with exponential mixing rate.

• April 14, Thurs. 4-5pm, 140 WCharlton, Colloquium. Renming Song, UIUC.

Potential Theory of Subordinate Brownian Motions

A subordinate Brownian motion can be obtained by replacing the time parameter of a Brownian motion by an independent increasing Lévy process (i.e., a subordinator). Subordinate Brownian motions form a large subclass of Lévy processes and they are very important in various applications. In this talk, I will give a survey of some of the recent results in the study of the potential theory of subordinate Brownian motions. In particular, I will present recent results on sharp two-sided estimates on the transition densities of killed subordinate Brownian motions in smooth open sets, or equivalently, sharp two-sided estimates on the Dirichlet heat kernels of the generators of subordinate Brownian motions.

• Apr 19. Na Zhang.

On the Law of Large Numbers for Discrete Fourier Transform