This semester, we are meeting on Mondays 3:35-4:30pm in the Seminar Room 608, 2925 Campus Green Drive. Please email yizao.wang@uc.edu
if there is any questions.
- Aug 26, Jacek Wesolowski, Warsaw University of Technology, Poland,
Asymptotics of the overflow in urn models.
Consider a number, finite or not, of urns each with the same
fixed capacity $r$. Balls, arriving one after another, are randomly
distributed among the urns. An overflow is the number of balls which are
assigned to urns that already are full.
When $r = 1$, using analytic methods, Hwang and Janson (2008) gave
conditions under which the overflow (which in this case is just the
number of balls landing in non–empty urns) has an asymptotically Poisson
distribution as the number of balls grows to infinity.
We study asymptotics of the overflow in general situation, i.e. for
arbitrary $r$. In particular, we provide sufficient conditions for both
Poissonian and normal asymptotics for general $r$, thus extending Hwang–Janson’s work. Our approach relies on purely probabilistic
methods.
This is based on the joint work with Raul Gouet (Universidad de Chile,
Santiago) and Pawel Hitczenko (NSF, Washington).
- Sep 2, Labor Day, no talk.
- Wed Sep 4, 3:35pm, Room 608, 2925 CGD, Zaoli Chen, Cornell University, Extremes of subexponential long memory sequences.
Extreme value theory considers the extremal behavior of stochastic processes. For stationary sequences, both marginal tails and dependence structures matter in the extremal limit theorems. We will focus mainly on stationary sequences with subexponential tails in the talk, starting from power-law tails and moving towards lighter tails. I will exhibit some proved results on extreme limit theorems. Those results indicate the interesting interplay between tails and memories.
- Sep 9, Rajinder Mavi. Exponential random graphs.
Exponentially weighted random graphs (ERGs) are a generalization of traditional random graphs where the distribution is exponentially weighted by graph parameters. In the last decade, ERG have been finding extensive applications in the social and biological sciences. Traditionally, ERGs are specified as distributions for dense (small world) graphs. We will discuss the basic properties of ERGs in this context, including the large graph (graphon) limit and approximation by stochastic block models. We will also discuss interesting possible extensions to the sparse regime where much of the theory remains to be developed.
- Sep 16, Joseph Najnudel, University of Bristol, UK. Gaussian multiplicative chaos and random matrix theory.
We identify an equality between two objects arising from different contexts of mathematical physics: Kahane's Gaussian Multiplicative Chaos on the circle, and the Circular Beta Ensemble from Random Matrix Theory. This is obtained via an analysis of related random orthogonal polynomials, making the approach spectral in nature.
- Sep 23, Joseph Najnudel, University of Bristol, UK. Gaussian multiplicative chaos and random matrix theory, continued.
- Sep 30, Joseph Najnudel, University of Bristol, UK. On uniqueness in law of solutions of some SDE's defined on left-open intervals.
A large class of SDE's defined on a left-closed interval of time satisfies existence and uniqueness properties when the distribution of the initial value of the solution is fixed. When the interval of time is left-open, there is no initial value so the question of existence and uniqueness of solutions is more delicate. In this talk, we study two examples where we have existence and uniqueness in law by only assuming some general properties of regularity of the solutions at the left endpoint of the interval. One example is the Ornstein-Uhlenbeck process on the whole real line, the other one is a SDE which is involved in our paper with Chhaibi on the identity in law relating the Gaussian Multiplicative Chaos and the Circular Beta Ensemble.
- Oct 7, No talk.
- Oct 14, Magda Peligrad, Central limit theorems for Markov processes.
- Oct 21, Wlodek Bryc, Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality.
This talk is based on a forthcoming paper with Raouf Fakhfakh and Wojciech Mlotkowski. The topic of this paper are properties of variance functions of Cauchy-Stieltjes Kernel families generated by a compactly supported (standardized) probability measure. After a brief introduction, I will describe some algebraic operations that can be used to construct additional variance functions from known variance functions. I will describe all quadratic and all cubic variance functions. I will also show how Cauchy-Stieltjes Kernel families with polynomial variance functions are related to generalized orthogonality of some families of monic polynomials.
Slides
- Oct 28, Ju-Yi Yen.
- Dec 5, Thurs, 4-5pm, Wenpin Tang, UC Berkeley, Discrete and continuous ranking models.
In this talk, I will discuss two different 'ranking' models:
Mallows' ranking model and rank-dependent diffusions. In the first part,
I will discuss the rank-dependent diffusions. I will focus on two models:
Up the River model, and N-player games with fuel constraints. These
problems require treating carefully the corresponding PDEs. The former
is joint with Li-Cheng Tsai, and the latter joint with Xin Guo and Renyuan Xu.
In the second part, I will focus on the Mallows' permutation, and various
generalizations. In particular, I will talk about a general model, called
regenerative permutations. I will also discuss the statistical properties and
algorithms of these Mallows' type ranking models. This is partly joint with
Jim Pitman.
If time permits, I will discuss recent progress on the random walk derived
from random permutations, which is motivated by applications in systems biology.
- Dec 6, Fri, 4-5pm, Helmuth Tyler, University of Bristol, Recurrence of the vertex-reinforced jump process in two dimensions.
Linearly-reinforced random walks have a preference to revisit previously visited locations. These models were introduced by Persi Diaconis to model how he walked around the streets of Paris in the 1980s. This playful invention turned out to be quite inspired — the highly non-Markovian way in which the history of the walk affects the future trajectory leads to interesting probabilistic challenges. Moreover, research over the past decade has revealed deep connections between linearly reinforced random walks and the mathematics of disordered electron systems (i.e., Anderson localization). I’ll introduce these topics and describe some of the connections. In particular, I will focus on the vertex-reinforced jump process, and I will highlight how these connections can be used to prove this process is recurrent on Z^2 for all values of the reinforcement strength.
Based on joint work with Roland Bauerschmidt and Andrew Swan.
- Dec 9, Mon, 4-5pm, Xiaoqin Guo, University of Wisconsin-Madison, Random walks and stochastic homogenization in a balanced random environment.
Stochastic homogenization studies the effective equations or laws that characterize the large scale phenomena for systems with complicated random dynamics at microscopic levels. In this talk, we explore the relation between stochastic homogenization and a probabilistic model called random motion in a random medium. In particular we focus on dynamics on the integer lattice which is non-reversible in time and defined by a non-divergence form difference operator. We will present qualitative and quantitative results for the diffusive behavior of the random walks and the homogenization of the corresponding random difference operator.
- Dec 10, Tues, 2:30-3:30pm, Chris Janjigian, University of Utah,
Stationary models in the Kardar-Parisi-Zhang universality class.
This talk will consider one of the most active areas of research in probability theory, the Kardar-Parisi-Zhang (KPZ) universality class. We will discuss what universality and stationarity mean in this context, along with what types of physical phenomena the KPZ class is conjectured to describe. After these generalities, we will discuss the expected relationship between stationary models and the universality conjectures along with some recent progress.
Past Seminar Archives (with photos!!)
Past Probability Events at UC
Other Probability Conferences
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