Spring 2012
Wednesdays 1.10 -2.00 pm in Carver 282
Organizer: Ananda Weerasinghe
Schedule and Abstracts
Wednesday, January 18,
Ananda Weerasinghe
Title: Introduction to the theory of large deviations- I.
Wednesday, January 25,
Ananda Weerasinghe
Title: Introduction to the theory of large deviations- II.
Wednesday, February 1,
Alexander Roitershtein
Title: Sanov's Theorem in the theory of large deviations.
This talk will be independent of the previous talks on large deviations.
Wednesday, February 8,
Alexander Roitershtein
Title: Sanov's Theorem in the theory of large deviations.
The proof of Sanov's theorem will be discussed during this talk.
Wednesday, February 15,
Alexander Roitershtein
Title: Sanov's Theorem in the theory of large deviations.
After a brief overview of last week's discussion, proof of Sanov's theorem will be completed in this talk.
Wednesday, February 22,
Ryan Martin
Title: The Lovaz Local Lemma
Abstract: The Lovasz local lemma (which, despite the alliteration, is due to both Erdos and Lovasz) is a way of measuring the probability of the
intersection of events among whom there are very few dependencies. This is a very useful and now classical tool for solving a wide variety
of combinatorial problems. I will state the lemma and will either prove it or do an application, depending on the interest
of the audience and the mood of the speaker.
Wednesday, February 29,
Youngsoo Seol
Title: Random Walk in a Sparse Random Environment.
Abstract: We will discuss the theory of random walk in a random environment(RWRE) on the integer lattice Z.
Then we will introduce a random walk in a sparse random environment(RWSE) on Z and some recent results. This research is related to Ph.D. thesis of the speaker written under the supervision of
Alex Roitershtein and Anastasios Matzavinos.
Wednesday, March 7,
Krishna B. Athreya
Title: The past of a Markov Chain.
Abstract: Let (X_n) be a discrete-time Markov Chain. Given the present value of X_n, we look at the past values X_(n-1), X_(n-2), ….. It will be shown that if the Markov chain is irreducible, aperiodic and positive recurrent, then
for each i and k, (X_(n-1), X_(n-2), X_(n-3), ….X_(n-k) | X_n=i ) converges in distribution to (Y_1, Y_2, Y_3, ….Y_k | Y_0=i) where (Y_i) is a Markov chain. Extensions to null-recurrent Markov Chains and continuous-time Markov Chains will also be discussed.
Monday, March 19,
Jonathan Peterson
Purdue University
Title: Large deviations and slowdown asymptotic for excited random walks.
Abstract: Excited random walks (also called cookie random walks) are self-interacting random walks where the transition probabilities depend on the number of previous visits to the current location. Although the models are quite different, many of the known results for one-dimensional excited random walks have turned out to be remarkably similar to the corresponding results for random walks in random environments. For instance, one can have transience with sub-linear speed and limiting distributions that are non-Gaussian. In this talk I will prove a large deviation principle for excited random walks. The main tool used will be what is known as the "backwards branching process" associated with the excited random walk, thus reducing the problem to proving a large deviation principle for the empirical mean of a Markov chain (a much simpler task). While we do not obtain an explicit formula for the large deviation rate function, we will be able to give a good qualitative description of the rate function. While many features of the rate function are similar to the corresponding rate function for RWRE, there are some interesting differences that highlight the major difference between RWRE and excited random walks.
Friday, March 30,
Jyy-I(Joy) Hong
Title: Coalescence on Multi-type Branching Processes.
Abstract: Consider a d-type Galton-Watson branching process. Pick k>1 individuals at random from the nth generation by simple randomsampling without replacement. Trace their lines of descent backward in time till they meet. Let X_{n,k} be the generation number of the coalescence time of these k individuals. We call the common ancestor of these chosen individuals in the X_{n,k}th generation their last common ancestor. Also, let Y_n be the number of the generation which the last common ancestor of the entire population in the nth generation. The limit behaviors of the distributions of X_{n,k} andY_n and some related properties will be discussed for supercritical, critical and subcritical cases.
Fall 2011
Mondays, Carver 401, 2:10-3:00 PM
Organizers: Krishna Athreya and Ananda Weerasinghe
Monday, August 29,
Alexander Roitershtein
Title: On a directionally reinforced random walk.
Abstract: We consider a slightly generalized version of a directionally reinforced random walk, which was originally introduced by Mauldin, Monticino, and von Weizsacker. Our main result is a stable limit theorem for the motion in higher dimensions. This extends a result of Horvath and Shao that was previously obtained in dimension one only. This is a joint work with Arka Ghosh and Reza Rastegar.
Monday, September 12,
Wolfgang Kliemann
Title: Dynamical systems with random perturbations.
Abstract: We will look at the situation of continuous time - continuous state space dynamical systems that are perturbed by a diffusion (Markov) process. We will distinguish two cases: the singular situation in which limit sets of the dynamical system remain unchanged under the perturbation, and the regular situation in which limit sets change with the perturbation. In the singular situation we will present some ideas for the local study of the perturbed system, including Lyapunov exponents, invariant manifolds, and Grobman-Hartman type of results. In the regular situation we will assume hypoellipticity of the system generator, and present some results on invariant measures and ergodicity. Applications include some 'large scale' electric power systems with random loads and random transfer coefficients.
Monday, September 19,
Krishna B. Athreya
Title: Introduction to Branching Processes
Abstract: This will be an expository (not a research) talk on discrete time, single type branching processes. The trichotomy based on the offspring mean(m) and the basic limit theorems for m<, =, > 1 will be discussed. If time permits, recent results on coalescence will be described.
Monday, September 26,
Hans Shuh, University of Mainz, Germany
Title: A Galton-Watson Branching Process with a threshold
Abstract: The extinction time of a Galton-Watson process which is supercritical below and subcritical above a certain fixed threshold, will be discussed.
Monday, October 3
Reza Rasteger
Title: Maximum occupation time of a transient excited random walk on Z.
Abstract: An excited random walk on Z is a modification of the nearest neighbor simple random walk such that in several first visits to each site of the integer lattice, the walk's jump kernel gives a preference to a certain direction. If the current location of the random walk has been already visited more than a certain number of times, then the walk moves to one of its nearest neighbors with equal probabilities.In this talk, we consider a transient excited random walk on Z and study the asymptotic behavior of the occupation time of a currently most visited site. This is a joint work with Alex Roitershtein.
Monday, October 10
Shubhro Bhattacharya
Title: Stock loans under bankruptcy I
Abstract: We address the issue of valuation of a financial derivative known as "Stock Loan" when the underlying asset is subject to bankruptcy. We use two types of bankruptcy models from the credit risk literature in our work.
Monday, October 17
Shubhro Bhattacharya
Title: Stock loans under bankruptcy II
Abstract: We address the issue of valuation of a financial derivative known as "Stock Loan" when the underlying asset is subject to bankruptcy. In this talk, we use a reduced form model from the credit risk literature to model bankruptcy.
Monday, October 24 (joint Probability/ Math Biology semiar)
Amy Ekanayake, Dept. of Math., Western Illinois University
Title: Stochastic SIS Epidemic Models for Metapopulations and their Comparison
Abstract:
A spatially explicit deterministic model will be derived for a susceptible-infected-susceptible (SIS) epidemic spreading through a metapopulation. Two types of stochastic modeling formats are then developed based on the assumptions of the deterministic model: a continuous time Markov chain (CTMC) model and an Itô stochastic differential equation (SDE) model. CTMC models, which have discrete state space, are common in population biology. Recently, Itô SDEs, with continuous state space, have also been applied. SDE models have advantages over CTMC models, in that numerical simulations of SDE models are generally much faster, especially for large population sizes. A comparison of numerical results for the stochastic models shows close agreement between CTMC and SDE models. As statistical measures, the moments of the distributions are sought for comparison. The differential equations for the moments of the distributions associated with the two models are not closed; each moment differential equation depends on higher-order moments. Various moment closure assumptions for metapopulation models have been discussed in literature, but their usefulness for spatially explicit metapopulation models have yet to be explored. The appropriateness of common closure approximations is discussed.
Monday, October 31
Tiefeng Jiang, School of Statistics, Univ. of Minnesota
Title: Moments of Traces for Circular Beta Ensembles
Abstract: Let x_1, ..., x_n be random variables from Dyson's circular beta-ensemble with probability density function prod_{1 leq j< k leq n} |e^{i x_j} - e^{i x_k}|^{beta}. For each n > 1 and beta>0, we obtain inequalities on expectations of p_{mu}(Z_n), where Z_n=(e^{i x_1}, \cdots, e^{i x_n}) and p_{mu} is the power-sum symmetric function for partition mu. When beta=2, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have limit results on the moments. These results apply to the three important ensembles: Circular Orthogonal Ensemble (beta=1), Circular Unitary Ensemble (beta=2) and Circular Symplectic Ensemble (beta=4). The main tool is the Jack function. This is a joint work with Sho Matsumoto.
Monday, November 7
Ryan Martin
Title: Random partitions of regular pairs.
Abstract:
In this talk, we will discuss epsilon-regular pairs, which are an important part of extremal graph theory and an elemental part of regularity theory and Szemeredi's regularity lemma. As a part of an ongoing project, we need an important lemma which states that large enough random subpairs of regular pairs are still regular.
We will discuss the lemma and how it is used in the more general results. Another piece of this puzzle will be discussed in the subsequent discrete math seminar on Nov. 08.
Tuesday, November 15, 4.10-5.00pm in Carver 268 (Please make note of the change of the day and time)
Chihoon Lee, Colorado State University
Title: Some stability properties of the reflected fractional Brownian motion on the positive orthant.
Abstract: This is the department colloquium talk for this week and please see the dept. colloquium calendar.