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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.

Series: Stochastics Seminar

The goal of this talk is to present a new method for sparse estimation
which does not use standard techniques such as $\ell_1$ penalization.
First, we introduce a new setup for aggregation which bears strong links
with generalized linear models and thus encompasses various response
models such as Gaussian regression and binary classification. Second, by
combining maximum likelihood estimators using exponential weights we
derive a new procedure for sparse estimations which satisfies exact
oracle inequalities with the desired remainder term. Even though the
procedure is simple, its implementation is not straightforward but it
can be approximated using the Metropolis algorithm which results in a
stochastic greedy algorithm and performs surprisingly well in a
simulated problem of sparse recovery.

Series: Stochastics Seminar

We propose a penalized orthogonal-components regression
(POCRE) for large p small n data. Orthogonal components are sequentially
constructed to maximize, upon standardization, their correlation to the
response residuals. A new penalization framework, implemented via
empirical Bayes thresholding, is presented to effectively identify
sparse predictors of each component. POCRE is computationally efficient
owing to its sequential construction of leading sparse principal
components. In addition, such construction offers other properties such
as grouping highly correlated predictors and allowing for collinear or
nearly collinear predictors. With multivariate responses, POCRE can
construct common components and thus build up latent-variable models for
large p small n data. This is an joint work with Yanzhu Lin and Min Zhang

Series: Stochastics Seminar

It is of interest that researchers study competing risks in which subjects may fail from any one of k causes. Comparing any two competing risks with covariate effects is very important in medical studies. In this talk, we develop omnibus tests for comparing cause-specific hazard rates and cumulative incidence functions at specified covariate levels. The omnibus tests are derived under the additive risk model by a weighted difference of estimates of cumulative cause-specific hazard rates. Simultaneous confidence bands for the difference of two conditional cumulative incidence functions are also constructed. A simulation procedure is used to sample from the null distribution of the test process in which the graphical and numerical techniques are used to detect the significant difference in the risks. In addition, we conduct a simulation study, and the simulation result shows that the proposed procedure has a good finite sample performance. A melanoma data set in clinical trial is used for the purpose of illustration.