"Fractional Brownian Motion in a multivariate setting"

This talk is about Fractional Brownian Motion (FBM), with a focus on the multivariate setting. We assume no background on the subject. FBM is a univariate Gaussian, self-similar, stationary-increment stochastic process. We begin with a discussion of some basic properties of FBM and of the relation between FBM and Long Range Dependent time series. Next, we talk about the extension of FBM to the multivariate setting, the so-called Operator Fractional Brownian Motion (OFBM). We show in what ways the theory for OFBM differs from that for FBM, and establish integral representations of OFBM in the spectral and time domains. We also study some basic properties of OFBM such as time-reversibility and the behavior of the spectral density around zero.

"Fractional Brownian Motion in a multivariate setting (part II)"

This talk is about Fractional Brownian Motion (FBM), with a focus on the multivariate setting. We assume no background on the subject. Last time we discussed some basic properties of FBM and of the relation between FBM and Long Range Dependent time series. This time we will talk about the extension of FBM to the multivariate setting, the so-called Operator Fractional Brownian Motion (OFBM). We show in what ways the theory for OFBM differs from that for FBM, and establish integral representations of OFBM in the spectral and time domains. If time allows, we also study some basic properties of OFBM such as time-reversibility and the behavior of the spectral density around zero.

"Sample Paths of Processes and Fields"

"Sample Paths of Processes and Fields (Part II)"

This is the second half of the topic: sample path properties. We will continue on modulus of continuity of local times with more details.  Then we will turn to the level set, which is connected to local times in a natural way. In fact, for Brownian motion the correct Hausdorffmeasure function of the level set is exactly the local time at the same point. If time permits, I well also present a very latest breakthrough(just this month, March) on the Hausdorff dimension of multiple points virtually for each and every process or field in the probability literature.

"Random Bits of Noise"

This talk is about Shannon Information, entropy and the capacity of a channel. A channel is a mechanism for transmitting information, and an important concept is the capacity of a channel. The main result of the talk gives a simple way to understand the capacity of a channel using Calculus. It goes on to describe generalizations of channels having a finite set of inputs and outputs, to ones that have infinite input- and output sets. This will touch on topological groups and semigroups, and their structure. They generalize the structures used in the case of finite inputs and outputs, which are the monoid of n-dimensional stochastic matrices and the submonoid of n-dimensional, doubly stochastic matrices. The talk will be introductory and accessible to students, providing the background necessary to understand the results