### Probability Seminar at University of CincinnatiFall 2015

This semester the regular meeting time of our seminar is Wed 3:35-4:30pm 4206 French Hall (Seminar Room). Please email yizao.wang@uc.edu if you want to be added to our email list or have any questions.
• Aug 26, Wed. From infinite urn schemes to decompositions of self-similar Gaussian processes. Yizao Wang.

We investigate a special case of infinite urn schemes first considered by Karlin (1967), especially its occupancy and odd-occupancy processes. We first propose a natural randomization of these two processes and their decompositions. We then establish functional central limit theorems, showing that each randomized process and its components converge jointly to a decomposition of certain self-similar Gaussian process. In particular, the randomized occupancy process and its components converge jointly to the decomposition of a time-changed Brownian motion $\mathbb B(t^\alpha),\alpha\in(0,1)$, and the randomized odd-occupancy process and its components converge jointly to a decomposition of fractional Brownian motion with Hurst index $H\in(0,1/2)$. The decomposition in the latter case is a special case of the decompositions of bi-fractional Brownian motions recently investigated by Lei and Nualart (2009). The randomized odd-occupancy process can also be viewed as correlated random walks, and in particular as a complement to the model recently introduced by Hammond and Sheffield (2013) as discrete analogues of fractional Brownian motions.

Joint work with Olivier Durieu.

• Sep 2, Wed. No seminar.

• Sep 9, Wed. No seminar.

• Sep 16, Wed. From infinite urn schemes to decompositions of self-similar Gaussian processes. Yizao Wang.

We will continue with the talk from last time, and talk about the proofs.

• Sep 23, Wed. No Seminar due to conflit with department colloquium.

• Sep 29, Tues, 4:00pm. Taft Lecture. Adventures in Mathematical Consulting. Larry Goldstein, Unviersity of Southern California.

Experience from four diverse consulting projects will be presented, ranging from the calculation of the winning odds in some simple and complicated sweepstakes giveaways to the physics of free falling luxury cars. Mathematical topics employed range from elementary probability, statistics, the birthday coincidence problem, size biasing, insurance risk calculations, linear regression, the Central Limit Theorem, Newtonian mechanics and the Stokes equation. Some lessons learned during the course of mathematical consulting will be summarized.

• Sep 30, Wed. Normal approximation for recovery of structured unknowns: Steining the Steiner formula, Larry Goldstein, University of Southern California.

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications. In particular, the discrete probability distribution ${\cal L}(V_{C})$ given by the sequence $v_0,\dots, v_d$ of conic intrinsic volumes of a closed convex cone $C$ in $\mathbb R^d$ summarizes key information about the success of convex programs used to solve for sparse vectors, and other structured unknowns such as low rank matrices, in high dimensional regularized inverse problems. The concentration of $V_C$ implies the existence of phase transitions for the probability of recovery of the unknown in the number of observations. Additional information about the probability of recovery success is provided by a normal approximation for $V_C$. Such central limit theorems can be shown by first considering the squared length $G_C$ of the projection of a Gaussian vector on the cone $C$. Applying a second order Poincaré inequality, proved using Stein’s method, then produces a non-asymptotic total variation bound to the normal for ${\cal L}(G_C)$. A conic version of the classical Steiner formula in convex geometry translates finite sample bounds and a normal limit for $G_C$ to that for $V_C$.

Joint with Ivan Nourdin and Giovanni Peccati. arxiv

• October 7, Wed. On the spectral density of stationary sequences and random fields. Magda Peligrad.

It is well-known that a stationary sequence or field with trivial tail sigma field has a spectral density. The situation is not so clear when a sequence of field which is a function of a double infinite i.i.d. (or regular) sequence. The talk will attempt to settle this question.

• October 8-10, Midwest Probability Colloquium at Northwestern University

• October 12, Mon, 5-6pm, Taft Lecture at Taft Research Center. Physics, Information, and Computation. Amir Dembo, Stanford University.

Theoretical models of disordered materials yield precise predictions about the efficiency of communication codes and the typical complexity of certain combinatorial optimization problems. The underlying common structure is that of many discrete variables, whose interaction is represented by a random 'tree like' sparse graph. We review recent progress in proving such predictions and the related algorithmic insights gained from it.

This talk is based on joint works with Andrea Montanari, Allan Sly, and Nike Sun.

• October 13, Tue, 2-3pm, Taft Research Center. Coloquium. Persistence Probabilities. Amir Dembo, Stanford University.

Persistence probabilities concern how likely it is that a stochastic process has a long excursion above fixed level and of what are the relevant scenarios for this behavior. Power law decay is expected in many cases of physical significance and the issue is to determine its power exponent parameter. I will survey recent progress in this direction (jointly with Sumit Mukherjee), focusing on stationary Gaussian processes that arise from the question how many algebraic polynomials of large even degree, with {-1,1}-valued coefficients, have no real zeros. Quite surprisingly, the same answer applies to having at the origin a positive solution for the heat equation initiated by white noise.

• October 23, Fri, 3:30-4:30pm 130 WCharlton. Colloquium. On static hedge of barrier options. Jiro Akahori, Ritsumeikan University, Japan.

I will introduce a new framework of static hedge of barrier options, which is based on an asymptotic expansion of heat kernel with Dirichlet boundary condition. I will also comment on a connection with robust hedging by Hobson and Obloj.

• October 28, Wed. Potential theory for random walks on graphs and fractals. Daniel Kelleher.

We shall talk about how to construct random walks on fractals from their approximating graphs, and talk about the difficulties that arise when trying to prove the convergence of these random walks in when these fractals are not post-critically finite. The focus will be on recent results for the hexacarpet and barycentric subdivision graphs.

• November 4, Wed. Asymmetric Simple Exclusion Processes with open boundary and Quadratic Harnesses. Wlodek Bryc.

I will talk about a connection between quadratic harnesses and a finite state asymmetric exclusion process (ASEP) with open boundaries. The relation gives quick access to the large deviations principle for the total number of particles in the system, and "explains" some of the explicit formulas that were discovered in mid 90's by the physicists.

This is joint work with Jacek Wesolowski.

• November 11, Wed. No Seminar (Veteran's Day)

• November 20, Fri, 3:35-4:30pm, 115 WCharlton. Limiting random operators for the Circular Unitary Ensemble. Joseph Najnudel, Institut de Mathématiques de Toulouse.

It is known that a unitary matrix can be decomposed into a product of reflections, one for each dimension, and that the Haar measure on the unitary group pushes forward to independent uniform measures on the reflections. We consider the sequence of unitary matrices given by successive products of random reflections. In this coupling, we show that powers of the sequence of matrices converge in a suitable sense to a flow of operators which acts on a random vector space. The vector space has an explicit description as a subspace of the space of sequences of complex numbers. The eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed in law according to a sine-kernel point process. The eigenvectors of the matrices converge almost surely to vectors which are distributed in law as Gaussian random fields on a countable set. This flow gives the first example of a random operator with a spectrum distributed according to a sine-kernel point process which is naturally constructed from finite dimensional random matrix theory.

• November 25, Wed. No seminar (Thanksgiving)

• December 2, Wed. Lifschitz Tails for random Schrodinger operator with Bernoulli distributed potentials. Vita Borovyk.

We will discuss an elementary proof of Lifschitz tail behavior for random discrete Schrodinger operator with a Bernoulli-distributed potential. The main idea is to approximate the low eigenvalues of the operator by the energies of sine waves supported on the islands where the potential takes its lower value.

Joint work with Michael Bishop and Jan Wehr.