# Past Probability Seminars Spring 2003

## UW Math Probability Seminar Spring 2003

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.

Organized by [index.html Timo Sepp�l�inen ]

### Schedule and Abstracts

|| Thursday, January 30 || || * Mike Kouritzin,* University of Alberta || || * Nonlinear filtering, the homeomorphism, and weak convergence * ||

In this talk, we will motivate the need for general historical measure-valued weak convergence results through nonlinear filtering theory and simulation. Our main tool for establishing such a result is the homeomorphism between a Polish space and a precompact subset of R-infinity that was introduced by Bhatt and Karandikar in 1993 and is related to the Tychonoff homeomorphism used in the Stone-Cech compactification. In the process of sketching the proof of our historical measure-valued result we will slightly generalize the statements and simplify the proofs of a few familiar weak convergence results. The talk will be based upon joint, ongoing work with Professor Douglas Blount that resulted from industrial collaboration.

|| Thursday, February 6 || || * Pavel Bleher, * Indiana University-Purdue University Indianapolis || || * Correlations between zeros of random polynomials and their scaling limits * ||

We prove the existence of the scaling limit of zeros of random polynomials and we obtain explicit formulae for the correlations between zeros in the scaling limit. In the talk I will discuss results for different ensembles of random polynomials and their multidimensional generalizations. The correlation between zeros exhibits repulsion in dimension 1, neutrality in dimension 2, and attraction in higher dimensions. I will discuss also universality theorems for the scaling limit of zeros.

|| Thursday, February 13 || || * Bret Larget, * UW-Madison Statistics and Botany || || * A Markov chain Monte Carlo approach to the estimation of evolutionary relationships from genome arrangements * ||

Our Bayesian approach to the analysis of genome arrangement data uses probability, both to model the processes that rearrange genomes and to compute posterior probabilities via Markov chain Monte Carlo. Genome arrangement data has the potential to be especially informative about evolutionary relationships among distantly related taxa. In this talk I will describe our use of probability models, discuss their application to the genome arrangements of mitochondria in animal phyla, and present directions of current and future research.

|| Thursday, February 20 || || * Mike Kouritzin,* University of Alberta || || * Nonlinear filtering, the homeomorphism, and weak convergence * || || The talk from January 30 is continued. ||

|| Thursday, February 27 || || * Firas Rassoul-Agha, * Ohio State University || || * Limit theorems for random walks in random environments * ||

The approach of the point of view of the particle is used to prove a law of large numbers and a large deviations principle for multi-dimensional random walks in a mixing random environment.

|| Thursday, March 6 || || *Sergey Bobkov, * University of Minnesota || || * Concentration of distributions of normalized sums * ||

For non-correlated random variables X_1 ,..., X_n, we are considering normalized sums of the form S = (X_{i_1} + ... + X_{i_k}) / \sqrt{k} for all possible increasing sequences of indices of length k. We are discussing the following property: when k is large, distribution functions of such sums are strongly concentrated about certain typical distribution (which might be normal under further mild assumptions).

|| Thursday, March 13 || || * * || || * * ||

|| Thursday, March 20 || || SPRING BREAK ||

|| Thursday, March 27 || || * Nigel Boston, * UW Madison || || * Strategies for the weakest link * ||

On the television program "The Weakest Link", each correct answer raises the pot, whereas an incorrect answer sends the pot back to zero. Before the next question, the player chooses either to bank or to risk the pot. We investigate how successful the strategy of banking after every r correct answers is, both theoretically and with data from the show.

|| Thursday, April 3 || || * * || || * * ||

|| Thursday, April 10 || || * * || || * * ||

|| Thursday, April 17 || || *Thomas G. Kurtz, * UW Madison || || * Particle representations for measure-valued branching processes * ||

A class of particle systems is considered in which each particle has a "type" or "location" in a state space E and a nonnegative "level." At each time t, the point process given by the locations and levels is conditionally a Poisson process with conditional mean measure given by the product of a random measure K(t) on E and Lebesgue measure on the space of levels. The locations of the particles evolve as independent Markov processes and each level evolves as a solution of an ordinary differential equation whose coefficients can depend on the particle's location. Each particle gives birth to new particles. The initial location of a new particle is that of the parent and the initial level lies above that of the parent. A judicious choice of rates and coefficients gives a model for which K(t) is a Dawson-Watanabe measure-valued branching process. The construction provides a tool for understanding the behavior of the Dawson-Watanabe process and related measure-valued processes.

|| Thursday, April 24 || || * Wei Sun, * University of Alberta || || * Branching Particle Method for Nonlinear Filtering * ||

Filtering is a method of estimating the conditional probability distribution of a random dynamic signal based upon a distorted, corrupted, partial sequence of observations. Particle approximations have the desirable feature of replacing storage of conditional densities for the filtering equations with particle locations.

We will first give an overview about the recently developed branching particle method for constructing computer workable filtering algorithms. In this method, a particle system with observation-dependent branching starts with n particles. Its empirical measure at time t, \mu _t^n, then closely approximates the unnormalized conditional distribution \mu_t of the signal to be estimated based upon the back observations. The branching mechanism is cautious in the sense that it does not over-resample but rather only kill or duplicate particles if the observations suggest that this is absolutely necessary. We compare this method with the earlier non-cautious interacting, branching methods introduced by Del Moral, Crisan, and Lyons and discuss its use in filtering reflecting diffusions in random environments.

In the second part of this talk, we will analyze the convergence rate of this branching method. The branching rate is scaled down as the observation noise magnitude \varepsilon \searrow 0, which reduces computational complexity as well as the probability that the number of particles will die down or explode quickly as compared to methods where each particle branches at every observation time regardless of the value of \varepsilon . We use a symmetric stable signal model and a standard Sobolev space to analyze our algorithm on.

|| Thursday, May 1 || || * Chris Hoffman,* University of Washington || || * Mixing Time for Biased Card Shuffling * ||

Consider a deck of n cards labeled 1 through n. We employ the following biased shuffling. At each stage we pick a pair of adjacent cards uniformly at random. Then with probability p>.5 we replace the cards with the lower numbered card before the higher numbered card. With probability 1-p we replace the cards with the higher numbered card first. We prove that the mixing time for this system is O(n^2). This proves a conjecture of Diaconis and Ram. This is joint work with Itai Benjamini, Noam Berger, and Elchanan Mossel.