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The CUNY Probability Seminar is typically held on Tuesdays at 4pm in the CUNY Graduate Math Department. The exact dates, times and locations are mentioned below.

Seminar List for Spring 2007

Feb 6, 2007 4:00pm, Room 5417click for b/w(color) postscript file
Speaker Dan Romik, Bell Labs
Title Random square Young tableaux
Abstract An NxN square Young tableaux is an NxN matrix containing the numbers 1 through N2 such that each row and column are increasing. It can be thought of as a sequence of instructions for building an N-by-N-shaped wall made of N2 unit square bricks by laying one brick at a time so that at each point during the construction the wall is stable and will not collapse. In this talk, I will describe the problem of choosing a uniformly random square Young tableau when N is large, and how the typical shape profile of the wall can be found using a large deviations analysis. I will also describe some applications of this result, and connections to other important problems such as finding the length of the longest increasing subsequence in a random permutation.
Feb 13, 2007 4:00pm, Room 5417click for b/w(color) postscript file
Speaker Robin Pemantle, University of Pennsylvania
Title Quantum random walks in one and two dimensions
Abstract I start by defining and motivating a quantum version of the simple random walk. This object has been understood in one dimension for some time, but not in higher dimensions. I will give a complete analysis in one dimension. In two dimensions, I will describe results that are currently being written down.
Mar 6, 2007 4:00pm, Room 5417click for b/w(color) postscript file
Speaker Bo'az Klartag, Princeton University
Title A central limit theorem for convex sets.
Abstract Suppose X is a random vector, that is distributed uniformly in some n-dimensional convex set. It was conjectured that when the dimension n is very large, there exists a non-zero vector u, such that the distribution of the real random variable < X,u > is close to the gaussian distribution. A well-understood situation, is when X is distributed uniformly over the n-dimensional cube. In this case, < X,u > is approximately gaussian for, say, the vector u = (1,...,1) / sqrt(n), as follows from the classical central limit theorem.
We prove the conjecture for a general convex set. Moreover, when the expectation of X is zero, and the covariance of X is the identity matrix, we show that for 'most' unit vectors u, the random variable < X,u > is distributed approximately according to the gaussian law. We argue that convexity - and perhaps geometry in general - may replace the role of independence in certain aspects of the phenomenon represented by the central limit theorem.
Mar 20, 2007 4:00pm, Room 5417click for b/w(color) postscript file
Speaker Vladimir Dobric, Lehigh University
Title Fractional Brownian motion,its martingales and natural wavelets
Abstract We have constructed the whole family of fractional Brownian motions as a single Gaussian field indexed by time and the Hurst index simultaneously. That field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. In this Gaussian field the pairs (H,H') of Hurst indices with the property H+H'=1, which we call the dual pairs, are essential tools for constructing "natural" martingales associated with fractional Brownian motions. The existence of those martingales, via their stochastic representations, leads to "the natural wavelet expansions" of those processes in the spirit of our earlier work on construction of natural wavelets associated to Gaussian-Markov processes.
Apr 17, 2007 4:00pm, Room 5417click for b/w(color) postscript file
Speaker Joseph Yukich, Lehigh University
Title Limit theorems for convex hulls.
Abstract We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sample. The approach involves introducing a generalized spatial birth growth process allowing for cell overlap (joint with T. Schreiber).
Apr 24, 2007 4:00pm, Room 5417click for b/w(color) postscript file
Speaker Wenbo V. Li, University of Delaware
Title Spectral Analysis of Brownian Motion with Jump Boundary
Abstract Consider a family of probability measures {my : y Î D} on a bounded open domain D Ì Rd with smooth boundary. For any starting point x Î D, we run a a standard d-dimensional Brownian motion B(t) until it first exits D at time t, at which time it jumps to a point in the domain D according to the measure mB(t) and starts the Brownian motion afresh. The same evolution is repeated independently each time the process reaches the boundary. The resulting diffusion process is called Brownian motion with jump boundary (BMJ). The spectral gap of non-self-adjoin generator of BMJ, which describes the exponential rate of convergence to the invariant measure, is studied. The main analytic tool is Fourier transforms with only real zeros.
May 1, 2007 4:00pm, Room 5417click for b/w(color) postscript file
Speaker Antonia Földes, College of Staten Island, CUNY
Title TRANSIENT NEAREST NEIGHBOR RANDOM WALK ON THE LINE
Abstract Let X0=0, X1,X2,... be a Markov chain with

P{Xn+1=i+1 | Xn=i}
=
1- P{Xn+1=i-1 | Xn=i}=
=
ì
ï
í
ï
î
1/2   
if    i=0,
1/2+pi   
if    i=1,2,...,
1/2-pi   
if    i=-1,-2,...
This sequence {Xi} describes the motion of a particle which is going away from 0 with a larger probability than to the direction of 0. That is to say 0 has a repelling power which becomes small if the particle is far away from 0. We intend to characterize the motion {Xi} and its local time.
May 8, 2007 4:00pm, Room 5417click for b/w(color) postscript file
Speaker Elena Kosygina, CUNY, Baruch
Title On the positive speed for one-dimensional "cookie" random walks
Abstract We consider exited random walks in one dimension: put M "cookies" at each site of the one dimensional integer lattice and let a random walker start from the origin.
Whenever there is a "cookie" at the walker's present site he will ëat" one cookie and
make one step to the right or one step to the left with probabilities p > 1/2 or 1-p < 1/2
respectively. If all the "cookies" at the walker's current location are already eaten then
the walker chooses the next step with equal probabilities. A result of M. Zerner states
that such walk is transient if and only if M(2p-1) > 1. Very recently A.-L. Basdevant and
A. Singh have shown that the walk has positive speed if and only if M (2p-1) > 2. We shall
discuss the method and give a sketch of the proof of this result.
May 15, 2007 4:00pm, Room 5417click for b/w(color) postscript file
Speaker Kavita Ramanan, Carnegie Mellon University
Title ON SOLUTIONS TO A CLASS OF STOCHASTIC DIFFERENTIAL INCLUSIONS
Abstract We establish sufficient conditions for existence and pathwise uniqueness of strong solutions to a class of possibly degenerate stochastic differential equations with discontinuous and possibly singular drift (interpreted as stochastic differential inclusions). We also allow for possibility of reflection in a polyhedral domain with piecewise constant reflection field. As motivation for this work, we show how our results may be applied to obtain limit theorems for several classes of stochastic networks. This includes joint work with Rami Atar and Amarjit Budhiraja.