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Series: Stochastics Seminar

In this talk, we resolve several questions related to a certain heavy
traffic scaling regime (Halfin-Whitt) for parallel server queues, a family
of stochastic models which arise in the analysis of service systems. In
particular, we show that the steady-state queue length scales like
$O(\sqrt{n})$, and bound the large deviations behavior of the limiting
steady-state queue length. We prove that our bounds are tight for the
case of Poisson arrivals. We also derive the first non-trivial bounds for
the steady-state probability that an arriving customer has to wait for
service under this scaling. Our bounds are of a structural nature, hold
for all $n$ and all times $t \geq 0$, and have intuitive closed-form
representations as the suprema of certain natural processes. Our upper
and lower bounds also exhibit a certain duality relationship, and
exemplify a general methodology which may be useful for analyzing a
variety of stochastic models. The first part of the talk is joint work
with David Gamarnik.

Series: Stochastics Seminar

The Burgers equation is a nonlinear PDE and one of the basic
hydrodynamic models. The ergodic theory of the Burgers turbulence began with
the work of E, Khanin, Mazel, Sinai (Ann. Math. 2000). In their paper and in
subsequent papers by Khanin and his coauthors, the compact case (Burgers on
a circle or torus) was studied. In this talk, I will discuss the noncompact
case. The main object is optimal paths through clouds of Poissonian points.

Series: Stochastics Seminar

The Potts antiferromagnet on a random graph is a model
problem from disordered systems, statistical mechanics with random
Hamiltonians. Bayati, Gamarnik and Tetali showed that the free energy
exists in the thermodynamic limit, and demonstrated the applicability
of an interpolation method similar to one used by Guerra and
Toninelli, and Franz and Leone for spin glasses. With Contucci,
Dommers and Giardina, we applied interpolation to find one-sided
bounds for the free energy using the physicists' ``replica symmetric
ansatz.'' We also showed that for sufficiently high temperatures, this
ansatz is correct. I will describe these results and some open
questions which may also be susceptible to the interpolation method.

Series: Stochastics Seminar

Given a random word of size n whose letters are drawn independently from an ordered alphabet of size m,
the fluctuations of the shape of the associated random RSK Young tableaux are investigated, when n and m converge
together to infinity. If m does not grow too fast and if the draws are uniform, then the limiting shape is the
same as the limiting spectrum of the GUE. In the non-uniform case, a control of both highest probabilities
will ensure the convergence of the first row of the tableau, i.e. of the length
of the longest increasing subsequence of the word, towards the Tracy?Widom distribution.

Series: Stochastics Seminar

We consider the problem of combining a (possibly uncountably infinite) set of affine estimators
in non-parametric regression model with heteroscedastic Gaussian noise. Focusing onthe exponentially weighted aggregate, we prove a PAC-Bayesian type inequality that leads tosharp oracle inequalities in discrete but also in continuous settings. The framework is general
enough to cover the combinations of various procedures such as least square regression,kernel ridge regression, shrinking estimators and many other estimators used in the literatureon statistical inverse problems. As a consequence, we show that the proposed aggregate provides
an adaptive estimator in the exact minimax sense without neither discretizing the rangeof tuning parameters nor splitting the set of observations. We also illustrate numerically thegood performance achieved by the exponentially weighted aggregate. (This is a joint work with Arnak Dalalyan.)

Series: Stochastics Seminar

We consider weighted random ball model driven by a Poisson random measure on
\Bbb{R}^d\times \Bbb{R}^+\times \Bbb{R} with
product heavy tailed intensity and we are
interested in the functional describing the contribution of the model in
some configurations of \Bbb{R}^d.
The fluctuations of such functionals are
investigated under different types of scaling and the talk will discuss the
possible limits.
Such models arise in communication network to represent the transmission of
information emitted by stations
distributed according to the Poisson measure.

Series: Stochastics Seminar

Hosted by Christian Houdre and Liang Peng.

In this talk I will discuss random matrices that are
matricial analogs of the well known binomial, Poisson, and negative
binomial
random variables. The common thread is the conditional variance of X
given S = X+X', which is a quadratic polynomial in S and in the
univariate case describes
the family of six Meixner laws that will be described in the talk.
The Laplace transform of a general n by n Meixner matrix ensemble
satisfies a system of PDEs which is explicitly solvable for n = 2. The
solutions lead to a family of six non-trivial 2 by 2 Meixner matrix
ensembles. Constructions for the "elliptic cases" generalize to n by n
matrices.
The talk is based on joint work with Gerard Letac.

Series: Stochastics Seminar

We study four discrete time stochastic systems on $\bbN$ modelingprocesses of rumour spreading. The involved individuals can eitherhave an active ora passive role, speaking up or asking for the rumour. The appetite inspreading or hearing the rumour is represented by a set of randomvariables whose distributionsmay depend on the individuals. Our goal is to understand - based on those randomvariables distribution - whether the probability of having an infiniteset of individuals knowing the rumour is positive or not.

Series: Stochastics Seminar

We consider two random sequences of equal length n
and the alignments with gaps corresponding to their Longest
Common Subsequences. These alignments are called
optimal alignments. What are the properties of these
alignments? What are the proportion of different aligned
letter pairs? Are there concentration of measure
properties for these proportions? We will see that
the convex geometry of the asymptotic limit set of
empirical distributions seen along alignments can determine
the answer to the above questions.

Series: Stochastics Seminar

Semimartingales constitute the larges class of "good integrators" for which Ito
integral
could reasonably be defined and the stochastic analysis machinery applied.
In this talk we identify semimartingales within certain infinitely divisible processes.
Examples include stationary (but not independent) increment processes, such as fractional
and moving average
processes, as well as their mixtures. Such processes are non-Markovian, often possess long
range memory, and are of
interest as stochastic integrators. The talk is based on a joint work with Andreas
Basse-O'Connor.