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Series: Stochastics Seminar

Finding ground states of spin glasses, a model of disordered materials,
has a deep connection to many hard combinatorial optimization problems,
such as satisfiability, maxcut, graph-bipartitioning, and coloring.
Much insight has been gained for the combinatorial problems from the
intuitive approaches developed in physics (such as replica theory and
the cavity method), some of which have been proven rigorously recently.
I present a treasure trove of numerical data obtained with heuristic
methods that suggest a number conjectures, such as an equivalence
between maxcut and bipartitioning for r-regular graphs, a simple
relation for their optimal configurations as a function of degree r,
and anomalous extreme-value fluctuations in a variety of models, hotly
debated in physics currently. For some, such as those related to
finite-size effects, not even a physics theory exists, for others
theory exists that calls for rigorous methods.

Series: Stochastics Seminar

Recently functional data analysis has received considerable attention in
statistics research and a number of successful applications have been reported, but
there has been no results on the inference of the global shape of the mean regression
curve. In this paper, asymptotically simultaneous confidence band is obtained for the
mean trajectory curve based on sparse longitudinal data, using piecewise constant
spline estimation. Simulation experiments corroborate the asymptotic theory.

Series: Stochastics Seminar

We consider two problems: (1) estimate a normal mean under a general
divergence loss introduced in Amari (1982) and Cressie and Read
(1984) and (2) find a predictive density of a new observation drawn
independently of the sampled observations from a normal distribution
with the same mean but possibly with a different variance under the
same loss. The general divergence loss includes as special cases
both the Kullback-Leibler and Bhattacharyya-Hellinger losses. The
sample mean, which is a Bayes estimator of the population mean under
this loss and the improper uniform prior, is shown to be minimax in
any arbitrary dimension. A counterpart of this result for predictive
density is also proved in any arbitrary dimension. The admissibility of
these rules
holds in one dimension, and we conjecture that the result is true in
two dimensions as well. However, the general Baranchik (1970) class
of estimators, which includes the James-Stein estimator and the
Strawderman (1971) class of estimators, dominates the sample mean in
three or higher dimensions for the estimation problem. An analogous
class of predictive densities is defined and any member of this
class is shown to dominate the predictive density corresponding to a
uniform prior in three or higher dimensions. For the prediction
problem, in the special case of Kullback-Leibler loss, our results
complement to a certain extent some of the recent important work of
Komaki (2001) and George, Liang and Xu (2006). While our proposed
approach produces a general class of predictive densities (not necessarily
Bayes) dominating the predictive density under a uniform prior,
George et al. (2006) produced a class of Bayes
predictors achieving a similar dominance. We show also that various
modifications of the James-Stein estimator continue to dominate the
sample mean, and by the duality of the estimation and predictive
density results which we will show, similar results continue to hold
for the prediction problem as well.
This is a joint research with Professor Malay Ghosh and Dr. Victor Mergel.

Series: Stochastics Seminar

Let $S_n$ be a centered random walk with a finite variance, and define the new sequence
$\sum_{i=1}^n S_i$, which we call the {\it integrated random walk}. We are interested in
the asymptotics of $$p_N:=\P \Bigl \{ \min \limits_{1 \le k \le N} \sum_{i=1}^k S_i \ge
0 \Bigr \}$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$
is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other types of
random walks that include double-sided exponential and double-sided geometric walks (not
necessarily symmetric). We also prove that $p_N \le c N^{-1/4}$ for lattice walks and
upper exponential walks, i.e., walks such that $\mbox{Law} (S_1 | S_1>0)$ is an
exponential distribution.

Series: Stochastics Seminar

My aim is to explain how to prove multi-dimensional central limit
theorems for the spectral moments (of arbitrary degrees) associated with
random matrices with real-valued i.i.d. entries, satisfying some appropriate
moment conditions. The techniques I will use rely on a universality
principle for the Gaussian Wiener chaos as well as some combinatorial
estimates. Unlike other related results in the probabilistic literature, I
will not require that the law of the entries has a density with respect to
the Lebesgue measure.
The talk is based on a joint work with Giovanni Peccati, and use an
invariance principle obtained in a joint work with G. P. and Gesine
Reinert

Series: Stochastics Seminar

In this talk, we study an interacting particle system arising in the
context of series Jackson queueing networks. Using effectively nothing
more than the Cauchy-Binet identity, which is a standard tool in
random-matrix theory, we show that its transition probabilities can be
written as a signed sum of non-crossing probabilities. Thus, questions
on time-dependent queueing behavior are translated to questions on
non-crossing probabilities. To illustrate the use of this connection,
we prove that the relaxation time (i.e., the reciprocal of the
’spectral gap’) of a positive recurrent system equals the relaxation
time of a single M/M/1 queue with the same arrival and service rates as
the network’s bottleneck station. This resolves a 1985 conjecture from
Blanc on series Jackson networks.
Joint work with Jon Warren, University of Warwick.

Series: Stochastics Seminar

One of the most important stochastic partial differential equations,
known as the superprocess, arises as a limit in population dynamics.
There are several notions of uniqueness, but for many years only weak
uniqueness was known. For a certain range of parameters, Mytnik and
Perkins recently proved strong uniqueness. I will describe joint work
with Barlow, Mytnik and Perkins which proves nonuniqueness for the
parameters not included in Mytnik and Perkins' result. This
completely settles the question for strong uniqueness, but I will end
by giving some problems which are still open.

Series: Stochastics Seminar

The Black‐Scholes model for stock price as geometric Brownian motion, and the
corresponding European option pricing formula, are standard tools in mathematical
finance. In the late seventies, Cox and Ross developed a model for stock price based
on a stochastic differential equation with fractional diffusion coefficient. Unlike the
Black‐Scholes model, the model of Cox and Ross is not solvable in closed form, hence
there is no analogue of the Black‐Scholes formula in this context. In this talk, we
discuss a new method, based on Stratonovich integration, which yields explicitly
solvable arbitrage‐free models analogous to that of Cox and Ross. This method gives
rise to a generalized version of the Black‐Scholes partial differential equation. We
study solutions of this equation and a related ordinary differential equation.

Series: Stochastics Seminar

I will describe recent work on the behavior of solutions to
reaction diffusion equations (PDEs) when the coefficients in the
equation are random. The solutions behave like traveling waves moving
in a randomly varying environment. I will explain how one can obtain
limit theorems (Law of Large Numbers and CLT) for the motion of the
interface. The talk will be accessible to people without much knowledge
of PDE.

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 corresponding random RSK Young tableaux are investigated, when both

n and m converge together to infinity. If m does not grow too fast and

if the draws are uniform, 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 random word, towards the Tracy-Widom distribution.