### Probability Seminar at University of CincinnatiFall 2016

This semester the regular meeting time of our seminar is Mon 4:40-5:30pm, French Hall 4206. Please email yizao.wang@uc.edu if you want to be added to our email list or have any questions.
• Aug 22. Semester starts.

• Aug 29. Raouf Fakhfakh. Laboratory of Probability and Statistics, Faculty of Sciences, University of Sfax.

Free expectation of the inverse in a Cauchy-Stieltjes Kernel family

If the distribution of $X$ belongs to a Cauchy-Stieltjes Kernel family concentrated on the positive real line, we determine the expectation of the reciprocal of $X$ as a function of the expectation $m$ of $X$ and we characterize the cases where this function is an affine function of reciprocal of the mean, or a function of the reciprocal of an affine transformation of the mean.

• Sep 6 Tues, 2-3pm, Seminar Room. David Sivakoff, the Ohio State University.

Long-range neighborhood growth models

Neighborhood growth models, and in particular threshold growth models such as bootstrap percolation, have been well studied on lattices. In these models, an initial set of occupied vertices is enlarged by repeatedly adding vertices with sufficiently many occupied neighbors. The basic question is whether a randomly selected initially occupied set with density $p$ eventually leads to full occupation? For many lattices, such as $Z^d$, it is known that a sharp phase transition occurs in the final occupied set as the density $p$ varies. In contrast, we initiate the study of neighborhood growth with long-range (non-local) neighborhoods, and show that gradual phase transitions occur. We also study neighborhoods with both local and non-local directions, and show that in this setting one may observe sharp, gradual or hybrid phase transitions.

• Sep 16, Fri, 5-6pm, Seminar Room. Vladas Pipiras, UNC Chapel Hill.

Bivariate long-range dependent time series models with general phase

The focus of this talk is on bivariate (vector-valued) time series that exhibit long-range dependence (LRD) and, more specifically, on the so-called phase parameter, an important quantity that appears in the cross spectrum at the zero frequency and controls the asymmetry of the series at large time lags. Previously considered bivariate LRD models have necessarily special phase parameter values, and hence can be unsuitable to capture the general LRD behavior in bivariate time series. In this talk, I will introduce several bivariate LRD models that allow for general phase, including a bivariate extension of the celebrated FARIMA class with a proposed set of identifiable parameters. I will indicate their connections to bivariate counterparts of fractional Brownian motion, and raise several open problems. Finally, I will also discuss maximum likelihood inference for the proposed models, and present an application to the annualized US inflation rates for goods and services.

• Sep 22 Thurs, 4-5pm, 277 WCharlton. Department Colloquium. Jan Rosinski, University of Tennessee, Knoxville.

On the interlinks between Gaussian and Poissonian infinitely divisible processes

Gaussian and Poissonian infinitely divisible (ID) processes come from inherently different types of a stochastic noise, a continuous thermal noise and a discrete pulses noise, respectively. It is therefore surprising that the square of a Brownian motion is a Poissonian ID process, which is based on a discrete noise. We will further discuss this phenomenon as well as some related open problems.

The celebrated Cameron-Martin formula is one of the fundamental results for Gaussian processes. We propose an analogy of this formula for Poissonian ID processes, which can also be viewed as a perturbation result. When applied to ID squared Gaussian processes, this result gives the Dynkin's isomorphism theorem. The applicability of such identities relies on a precise understanding of Levy measures of stochastic processes.

• Sep 26. Yizao Wang.

Extremes of $q$-Ornstein-Uhlenbeck Processes

The $q$-Ornstein-Uhlenbeck processes, $q\in(−1,1)$, are a family of stationary Markov processes that converge weakly to the standard Ornstein-Uhlenbeck process as $q$ tends to 1. It has been noticed recently that in terms of path properties, however, for each $q$ fixed the $q$-Ornstein-Uhlenbeck process behaves qualitatively different from their Gaussian counterpart in several aspects. Here, two limit theorems on the extremes of $q$-Ornstein-Uhlenbeck processes are established. Both results are based on the weak convergence of the tangent process at the lower boundary, a positive self-similar Markov process little investigated so far in the literature. The first result is the asymptotic excursion probability established by the double-sum method, with an explicit formula for the Pickands constant in this context. The second result is a Brown-Resnick-type limit theorem on the minimum process of i.i.d. copies. With appropriate scalings in both time and magnitude, a new semi-min-stable process arises in the limit.

In this talk, I will focus on classical questions on extremes of Gaussian processes, in order to illustrate the methods applied in the study of $q$-Ornstein-Uhlenbeck processes.

• Oct 3. Yizao Wang.

I will continue from last week, and talk about multivariate extreme value theory and particularly max-stable processes. Most of the talk will be expository.

• Oct 10. Joseph Najnudel.

On the maximum of the characteristic polynomial of the Circular Beta Ensemble

In this talk, we present our result on the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the Circular Beta Ensemble. Using different techniques, it gives an improvement and a generalization of the previous recent results by Arguin, Belius, Bourgade on the one hand, and Paquette, Zeitouni on the other hand. They recently treated the CUE case, which corresponds to $\beta$ equal to 2.

• Oct 17. Wlodek Bryc.

Cauchy-Stieltjes kernel families with polynomial variance functions

This is a "progress report" on properties of Cauchy-Stieltjes Kernel that we managed to figure out during Raouf's visit to Cincinnati. The basic question is when a polynomial could serve as a "variance function" of a Cauchy-Kernel family. I will talk about connections to d-orthogonality, and give a positive and a negative example.

• Oct 27 Thurs 4-5pm, 277 WCharlton, Department Colloquium. Elizabeth Strouse, University of Bordeaux, France

Special matrices and measures, Eigenvalues and integrals

Toeplitz operators (with giant Toeplitz matrices) on $\ell^2(\mathbb N)$ can be interpreted as compositions of multiplications (by a 'symbol' function $f$) and orthogonal projections. The Szegö limit theorems describe a relationship between the spectrum of compressions of these operators to finite dimensional subspaces - the sequences of length $n$ - and the integral of the symbol. Recently a survey by Donald Sarason aroused much interest in 'truncated Toeplitz operators' on 'model spaces'. Model spaces are subspaces of $\ell^2(\mathbb N)$ which 'generalize' the finite sequence spaces; and truncated Toeplitz are generalizations of Toeplitz matrices. I will speak about these operators and Szegö-type theorems which hold for them.

• Nov 1, Tues, 12:30-1:30pm, Seminar Room, joint Analysis/FA/Probability Seminar. Elizabeth Strouse, Universit of Bordeaux, France

Different kinds of Hankel operators and different types of BMO

Functions of Bounded Mean Oscillation on the circle are those whose values do not 'oscillate' too much on intervals. It is a remarkable and deep result that these functions can be characterized by their Fourier coefficents and are related to boundedness of certain 'Hankel operators'. I will give an elementary explanation of these results and their various generalizations to functions of several variables.

• Nov 3, Thurs, 4-5pm, 277 WCharlton. Department Colloquium. Jan Wehr, University of Arizona.

Time scaling interaction in diffusive systems - experiment and mathematics

An experiment performed in 2011 in Naples studied a nonlinear electrical circuit in the presence of noise. In addition to the characteristic noise correlation time, another small time scale present in the system was the delay with which the system was responding to the noise. As both these times were varied in a controlled way, it was observed that the dynamics of the circuit was changing, according to a simple pattern. In a joint work with the authors of the experiment, we have explained this effect, deriving the effective stochastic differential equation, followed by the system in the limit of small delay and small noise correlation time. In view of their potential applications, the results of this work, in which two graduate students at the University of Arizona were also involved, were published in Nature Communications. Subsequently, a much more general result was proven, showing the pattern in which several small time scales compete and interact in diffusive systems. I am going to describe the experiment, introduce its mathematical model and present the theorem, putting it in the context of classical stochastic analysis. This is a joint work with Scott Hottovy, Austin McDaniel, Giuseppe Pesce and Giovanni Volpe. From a broad point of view, the effect studied in this work is an example of a noise-induced drift, also present in other diffusive systems.

• Nov 7. No meeting.

On the functional central limit theorem for non-stationary sequences

Several results will be presented concerning limit theorems for non-stationary sequences. They are based on martingale approximation with a view towards applications to the central limit theorem for Markov chains and evolution in a random time scenery.

• Nov 21. Ju-Yi Yen.

Excursion landscape

In this talk, we study the process obtained from a Brownian bridge after excising all the excursions below the waterline level which reach zero. Three variables of interest are the maximum of this process, the value where this maximum is attained, and the total length of the excursions which are excised. Our analysis relies on some interesting transformations connecting Brownian path fragments and the 3-dimensional Bessel process.

• Dec 2, Fri 4:40-5:30pm. 273 WCharlton. Zhipeng Liu, NYU.

TASEP on a ring in relaxation time scale

Gaussian fluctuation has been known as a universal law behind many mathematical physics models: quantities that are a large sum of i.i.d. random variables converge to Gaussian distribution, which is independent of the distribution of individual random variables. Recently a theory on a new universality class, the so-called KPZ universality class, has been rapidly developed. In this universality class, the one point distributions are expected to be universal and independent of the model, but only depend on the initial data.

In this talk, we consider a specific model which lies in the crossover between KPZ and Gaussian universality classes: the totally asymmetric simple exclusion process (TASEP) on a ring. It has been conjectured since the 80's that the system starts to relax when time is 3/2 power of the system size. Before the relaxation time scale, the system belongs to KPZ universality class and the limiting distribution depends on the initial data. And after the relaxation time scale, the system reaches equilibrium and the limiting distribution is Gaussian. We will show the crossover distributions in the relaxation time scale for three classical initial conditions. These crossover distributions interpolate the distributions in KPZ universality classes and the Gaussian distribution, and are expected to be universal for any periodic system in KPZ universality class.

• Dec 5. Exam week.