## CUNY Probability Seminar Spring 2002

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Speaker | Horng-Tzer Yau, Courant Institute | ||
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Title | Superdiffusivity of Two Dimensional Stochastic Dynamics | ||

Abstract | The asymmetric simple exclusion process (ASEP) is a system of
asymmetric random walks with hard core condition so that no two
particles can be on the same site. It has become the paradigm of
stochastic dynamics modeling the transport phenomena or the interface
growth. Furthermore, combining asymmetric simple exclusion and
suitable collision kernels, one can construct lattice gas models
converging to the incompressible Navier-Stokes equations in dimension
d = 3. In particular, the viscosity is finite. For d £ 2, the
diffusive behavior no longer holds. It was conjectured that its
diffusion coefficient diverges as t^{1/3} in d = 1 and
(logt)^{2/3} in d = 2 by Beijeren-Kutner-Spohn or the Kardar-Parisi-Zhang
equation. Recent results based on integrable systems has indicated
strongly the exponent 1/3 in the totally asymmetric case in
dimension d = 1. We shall describe an approach based on Euclidean
field theory proving this conjecture in dimension d = 2.
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Speaker | Victor Dela Pena , Columbia University | ||

Title | The LIL for self-normalised martingales | ||

Abstract | In this talk I introduce an extension of Stout's LIL to the
case of self-normalised martingales. The proof relies on the
construction of an exponential supermartingale involving
the martingale and the sum of its squared martingale differences.
This represents joint work with T. L. Lai and M. J. Klass | ||

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Speaker | Davar Khoshnevisan, University of Utah | ||

Title | Greedy Search on a Sparse Tree with Random Edge-Weights | ||

Abstract | I will discuss the asymptotic value of the greedy search on a sub-exponentially growing tree, when the edge-weights are i.i.d. random variables. Amongst other things, I will show the nonexistence of a üniversal algorithm" that can match greedy search. this is joint work with T. M. Lewis. | ||

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Speaker | Jan Rosi\'nski, University of Tennessee, Knoxville | ||

Title | Radonification of cylindrical semimartingales by a single Hilbert-Schmidt operator | ||

Abstract | It is well known that semimartingales constitute the most general basis for nonanticipating stochastic integration. An essential problem in the infinite dimensional stochastic integration is to describe the class of radonifying operators for cylindrical semimartingales. This problem was investigated by Badrikian and Ustunel (1996) who showed that a composition of three Hilbert-Schmidt operators radonifies a cylindrical semimartingale on a Hilbert space to a strong semimartingale. Analogous results regarding compositions of three operators were obtained by L. Schwartz (1994-96) in the context of Banach space valued semimartingales. We prove that just a single Hilbert-Schmidt operator suffices to radonify a cylindrical semimartingale on a Hilbert space. Since a Hilbert-Schmidt operator is necessary to radonify a cylindrical Brownian motion, this resolves the problem for Hilbert spaces. The proof relies on inequalities employing a Gaussian randomization. The talk will be based on a joint work with A. Jakubowski, S. Kwapien, and P. Raynaud de Fitte. | ||

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Speaker | Joseph Yukich, Lehigh University | ||

Title | Limit Theory for Random Packing | ||

Abstract | Consider sequential packing of unit balls in a large cube, as in the Renyi car-parking model, but in any dimension and with finite input. We prove a law of large numbers and central limit theorem for the number of packed balls in the thermodynamic limit. We prove analogous results for numerous related applied models, including cooperative sequential adsorption, ballistic deposition, and spatial birth-growth models. The proofs are based on a general law of large numbers and central limit theorem for ``stabilizing'' functionals of marked point processes of independent uniform points in a large cube, which are of independent interest. ``Stabilization'' means, loosely, that local modifications have only local effects. (based on joint work with Mathew Penrose). | ||

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Speaker | Kavita Ramanan, Bell Labs, Lucent Technologies | ||

Title | Large Deviations of Stationary Reflected Brownian Motion | ||

Abstract | We analyze the tail probabilities of stationary reflected Brownian motion in the N-dimensional nonnegative orthant having drift b, covariance matrix A and constraint matrix D. Under suitable stability and regularity conditions, the exponential decay rate of the tail probabilities (i.e. the rate function) is known to have a variational representation V(x). We discuss why this representation is hard to analyze, and develop new techniques for analyzing this problem. | ||

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Speaker | Richard Bass, University of Connecticut | ||

Title | Local times on curves for space-time Brownian motion | ||

Abstract | The local time for a curve f with respect to space-time Brownian motion measures the amount of time the Brownian motion spends in a band about f. I'll discuss some characterizations of these local times and then address the question of when the supremum of local times on curves over a large class of curves is finite or not. | ||

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Speaker | Evarist Giné, University of Connecticut, Storrs | ||

Title | On kernel density estimators: uniform convergence over the wholespace of the deviations from their means | ||

Abstract | Stute's law of the logarithm (rate of a.s. convergence) for the sup norm over compact sets of the deviation from the mean of a kernel density estimator, as well as the Bickel-Rosenblatt result on shift-convergence in distribution of the same quantities, are strengthened to the norm of the supremum over the whole space. Some of the tools used include Talagrand's exponential inequality for empirical processes, extension of classical results on stationary Gaussian processes to slightly not stationary and, of course, KMT. These results were obtained in collaboration with A. Guillou, and with V.I. Koltchinskii and L. Sachanenko. |