Probability Seminar List
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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 Fall 2009
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| Speaker |
Jean Bertoin, Universite Paris VI |
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| Title |
A limit theorem for the tree of alleles in branching processes with rare neutral mutations |
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| Abstract |
We are interested in the genealogical structure of alleles for a Bienayme-Galton-Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small.
We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families
converges in distribution to a certain continuous state branching process in discrete time. Ito's excursion theory and the Lévy-Itô decomposition of subordinators provide fundamental insights for the results. |
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| The author has provided the following link: | | | Additional lectures by Professor Bertoin in NYC |
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| Speaker |
Fredrik Johansson, KTH |
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| Title |
Optimal Holder exponent for the SLE path |
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| Abstract |
The Schramm-Loewner evolution (SLE) is a family of random fractal curves obtained by solving the Loewner equation with a Brownian motion input. SLE has attracted much attention in recent years since, for example, it can be used to rigorously understand scaling limits of several discrete models from statistical physics. In the talk we give a very brief introduction to SLE and discuss our proof of J. Lind's conjecture about the optimal Holder exponent for the SLE path parametrized by half-plane capacity.
This is joint work with G. F. Lawler (University of Chicago). |
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| Speaker |
David Mason, University of Delaware |
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| Title |
Large Self-normalized Levy process distributional behavior at small time and large time |
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| Abstract |
Let Xt be a Lévy process and
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Vt=s2t+ |
å
0 < s £ t
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(DXs)2, t > 0, |
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be its quadratic variation process, where DXt=Xt-Xt- denotes
the jump process of X. We give stability and compactness results at small
time, i.e, as t¯0, and large time, i.e, as t®¥ for
the self-normalized process Xt/Ö{Vt}. Special cases of our results characterize the possible limit
laws of Xt/Ö{Vt} at both small and large times. One such result
says that
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Xt/ |
Ö
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Vt
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\oversetD®N(0,1),\text ast¯0, |
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a standard normal random variable if and only if, for some nonstochastic
function b(t) > 0,
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Xt/b(t)\oversetD®N(0,1),\text as t¯0, |
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with the same statement holding as t®¥. Our asymptotic
normality results are the small time and large time self-normalized Lévy
process analogs of what is known for self-normalized sums of i.i.d. random
variables. It turns out that roughly speaking for small time behavior
everything is controlled by the tails of the Lévy measure of the process
near zero, and for large time behavior, it is determined by the tails of the
Lévy measure at infinity. Finally we show how stochastically compact
Lévy processes, both at zero and infinity, arise via subsequential
distributional limits of partial sums of i.i.d. random variables in the Feller
Class. This talk is based on ongoing joint work with Ross Maller of the
Australian National University. |
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| Speaker |
Ron Peled, Courant Institute, NYU |
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| Title |
Gravitational Allocation to Poisson Points |
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| Abstract |
One way to quantify how uniformly spread a point process is, is to allocate cells of equal volume to each of its points and measure the regularity of the resulting partition of space. Such allocations (also known as transportations, matchings or marriages) with an additional equivariance constraint, have been the subject of many investigations in recent years. I will survey results in the field, with special focus on a natural allocation rule - the Gravitational Allocation, and its quantitative geometry when used for the Poisson point process. |
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| Speaker |
Elena Kosygina, Baruch College, CUNY |
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| Title |
Limit laws of excited random walks on integers |
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| Abstract |
We consider excited random walks on integers with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the "cookies". E. Kosygina and M.P.W. Zerner have shown that when the total expected drift per site, d, is larger than 4 then the walks under the averaged measure obey the Central Limit Theorem. We show that when d Î (2,4] the limiting behavior of an appropriately centered and scaled excited random walk is described by a strictly stable law with parameter d/2. Our method also
extends the results obtained by A.-L. Basdevant and A. Singh for d Î (1,2] under the non-negativity assumption to the setting, which allows both positive and negative cookies.
(joint with T. Mountford) |
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| Speaker |
Dick Gundy, Rutgers University |
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| Title |
Tilings of R1, scaling functions, and a Markov process |
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| Abstract | TBA |
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| Speaker |
Hana Kogan, CUNY |
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| Title |
On infinite divisibility of Gaussian squares with non-zero mean |
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| Abstract |
I will present Feller's coefficient positivity condition for infinite divisibility and outline Griffith and Bapat's theorem on infinite divisibility of zero-mean Gaussian squares.
I will present the series expansion for the Laplace Transform in the non-zero mean case (content of my thesis) and give applications (in particular to the upper bound on their critical point). I will present the theorem on non-zero mean Gaussian squares divisibility, describe the significance of Ray-Knight Isomorphism theorem in the original proof and give an (elementary) alternative proof.
I will mention the unsolved questions connected to this topic.
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