OSU PROBABILITY SEMINARDepartment of Mathematics |
Day/Time/Room |
Speaker |
Title and abstract |
Tuesday, January 19, 4:00 pm WNGR 201 |
Bartłomiej Siudeja
University of Oregon |
"Transition densities and trace estimates for a broad class of Levy processes" Abstract. Transition density of a stochastic process allows one to quantify the dynamics of the process. Yet, except for Brownian motion and a very few special cases, there is no closed formula for the density, which is usually defined via a characteristic function. The problem is exacerbated for killed processes (confined to bounded domains), where even the Brownian case is not explicit. We will discuss recent progress on bounding the transition probabilities of a class of killed Levy processes using geometric properties of their domains. We will use these to estimate their traces, the quantities revealing so-called heat invariants. In the classical, Brownian case, the first two invariants are the volume and the surface area of the domain. Surprisingly, we will find the same simple quantities in traces of very general Levy processes. |
Tuesday, January 26, 4:00 pm WNGR 201 |
Albert M. Fisher
Department of Mathematics University of São Paulo, Brazil |
"The transition in renewal processes to fractal-like returns and ergodic infinite measures" Abstract. A renewal process counts the number of "events" in the case where the gaps between successive events are independent of each other and are distributed in the same way. One example is the number of burnt-out light bulbs up to a certain time; another is the number of returns to a given state of a countable state Markov process, for example the returns to zero of a random walk. Given the gap distribution, one can calculate the expected return time; there is a dichotomy between this (the first moment) being finite or infinite. One can make a finer distinction, considering the least moment alpha that is finite. Of particular importance is the second moment (the variance); whether or not this is finite subdivides the finite expectation region in two. Making the assumption of regularly varying tails, the result is three "phases" of asymptotic behavior: Gaussian, stable and Mittag-Leffler. From the point of ergodic theory, the renewal process counts the number of returns to a subset of finite measure, and as we transition through the three phases, we can observe the transition from finite to infinite invariant measure. Precisely, we show that in each case one has asymptotically self-similar returns, stated as an almost-sure invariance principle in log density. For the infinite measure case this is interpreted as fractal-like return structure, leading to an order-two ergodic theorem. |
Tuesday, February 9, 4:00 pm WNGR 201 |
Li Chen
Intel Corporation, Oregon |
"Discovering Insights from Graphs -- When Pattern Recognition Meets Social Network, Neural Connectomes and Digital Marketing" Abstract. A graph is a representation of a collection of interacting objects. The field of pattern recognition developed significantly in the 1960s, and the field of random graph inference has enjoyed much recent progress in both theory and application. This talk focuses on pattern recognition in the context of a particular family of random graphs, namely the stochastic blockmodels, from the two main perspectives of single graph inference and joint graph inference, as well as its applications in social network, neural connectomes and digital marketing. Single graph inference is the performance of statistical inference on one single observed graph. Given a single graph realized from a stochastic blockmodel, we here consider the specific exploitation tasks of vertex classification, clustering, and nomination. The theoretical guarantees of these methods are proved and their effectiveness are demonstrated in simulation as well as real datasets including communication network, online advertising, and neural connectomes. We are also concerned with joint graph inference, which involves the joint space of multiple graphs. Specifically, given two graphs, we consider the tasks of seeded graph matching for large graphs and joint vertex classification. The methodologies are shown to discover signals in the joint geometry of diffusion tensor MRI and the Caenorhabditis elegans neural connectomes. |
Tuesday, February 16, 4:00 pm WNGR 201 |
Huanqun Jiang
Oregon State University |
TBA Abstract. TBA |
Tuesday, March 1, 4:00 pm WNGR 201 |
José Javier Cerda Hernández
Oregon State University (visiting from Brazil) |
TBA Abstract. TBA |