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Series: Stochastics Seminar

We study a problem of estimation of a large Hermitian nonnegatively definite
matrix S of unit trace based on n independent measurements
Y_j = tr(SX_j ) + Z_j , j = 1, . . . , n,
where {X_j} are i.i.d. Hermitian matrices and {Z_j } are i.i.d. mean
zero random variables independent of {X_j}. Problems of this nature are
of interest in quantum state tomography, where S is an unknown density
matrix of a quantum system. The estimator is based on penalized least
squares method with
complexity penalty defined in terms of von Neumann entropy.
We derive oracle inequalities showing how the estimation error depends on the
accuracy of approximation of the unknown state S by low-rank matrices.
We will discuss these results as well as some of the tools used in their
proofs (such as generic chaining bounds for empirical processes and
noncommutative Bernstein type inequalities).

Series: Stochastics Seminar

The aim of this joint work with Ely Merzbach is to present a satisfactory definition of the class of set-indexedL\'evy processes including the set-indexed Brownian motion, the spatial Poisson process, spatial compound Poisson processesand some other stable processes and to study their properties. More precisely, the L\'evy processes are indexed by a quite general class $\mathcal{A}$ of closed subsets in a measure space $(\mathcal{T} ,m)$. In the specific case where $\mathcal{T}$ is the $d$-dimensional rectangle$[0 ,1]^d$ and $m$ is the Lebesgue measure, a special kind of this definition was given and studied by Bass and Pyke and by Adler and Feigin. However, in our framework the parameter set is more general and, it will be shown that no group structure is needed in order to define the increment stationarity property for L\'evy processes.

Series: Stochastics Seminar

We provide a sufficient condition for the continuity of real valued permanental processes. When applied to the subclass of permanental processes which consists of squares of Gaussian processes, we obtain the
sufficient condition for continuity which is also known to be
necessary. Using an isomorphism theorem of Eisenbaum and Kaspi
which relates Markov local times and permanental processes we obtain
a general sufficient condition for the joint continuity of the local
times.

Series: Stochastics Seminar

We consider various dependence concepts for random fields. Special attention is paid to Gaussian and shot-noise fields. The multivariate central limit theorems (CLT) are proved for the volumes of excursion sets of stationary quasi-associated random fields on $\mathbb{R}^d$. Formulae for the covariance matrix of the limiting distribution are provided. Statistical versions of the CLT are established as well. They employ three different estimators of the asymptotic covariance matrix. Some numerical results are also discussed.

Series: Stochastics Seminar

In this talk we shall discuss the problem of fitting a distribution function to the
marginal distribution of a long memory process. It is observed that unlike in the i.i.d.
set up, classical tests based on empirical process are relatively easy to implement.
More importantly, we discuss fitting the marginal distribution of the error process in
location, scale and linear regression models.
An interesting observation is that the first order difference between the residual
empirical process and the null model can not be used to asymptotically to distinguish
between the two marginal distributions that differ only in their means. This finding is
in sharp contrast to a recent claim of Chan and Ling to appear in the Ann. Statist.
that such a process has a Gaussian weak limit. We shall also proposes some tests based
on the second order difference in this case and analyze some of their properties.
Another interesting finding is that residual empirical process tests in the scale
problem are robust against not knowing the scale parameter.
The third finding is that in linear regression models with a non-zero intercept
parameter the first order difference between the empirical d.f. of residuals and the null
d.f. can not be used to fit an error d.f.
This talk is based on ongoing joint work with Donatas Surgailis.

Series: Stochastics Seminar

We consider a stochastic Navier-Stokes equation driven by a space-time Wiener process. This equation is quantized by transformation of the nonlinear term to the Wick product form. An interesting feature of this type of perturbation is that it preserves the mean dynamics: the expectation of the solution of the quantized Navier-Stokes equation solves the underlying deterministic Navier-Stokes equation. From the stand point of a statistician it means that the perturbed model is an unbiased random perturbation of the deterministic Navier-Stokes equation.The quantized equation is solved in the space of generalized stochastic processes using the Cameron-Martin version of the Wiener chaos expansion. A solution of the quantized version is unique if and only if the uniqueness property holds for the underlying deterministic Navier-Stokes equation. The generalized solution is obtained as an inverse of solutions to corresponding quantized equations. We will also demonstrate that it could be approximated by real (non-generalized processes). A solution of the quantized Navier-Stokes equation turns out to be nonanticipating and Markov. The talk is based on a joint work with R. Mikulevicius.

Series: Stochastics Seminar

The almost sure rate of convergence in the sup norm for linear wavelet density estimators is obtained, as well as a central limit theorem for the distribution functions based on these estimators. These results are then applied to show that the hard thresholding wavelet estimator of Donoho, Johnstone, Kerkyacharian and Picard (1995) is adaptive in sup norm to the smoothness of a density. An alternative adaptive estimator combining Lepski's method with Rademacher complexities will also be described. This is joint work with Richard Nickl.

Series: Stochastics Seminar

Abstract: We consider the hidden Markov model, where the dynamic of theprocess is modelled by a latent Markov chain Y and the observations X aresuch that: 1) given the realization of Y, the observations areindependent; 2) the distribution of the i-th observations (X_i) depends onthe i-th element of the Y (Y_i), only.The segmentation problem consists of estimating the underlying realization(path) of Y given the n observation. Usually the realization with maximumlikelihood, the so called Viterbi alignment is used. On the other hand, itis easy to see that the Viterbi alignment does not minimize the expectednumber of misclassification errors.We consider the segmentation problem in the framework of statisticallearning. This unified risk-based approach helps to analyse many existingalignments as well as defining many new ones. We also study theasymptotics of the risks and infinite alignments.

Series: Stochastics Seminar

The talk will present several
limit theorems for the supercritical colony of the particles with masses. Reaction-diffusion
equations responsible for the spatial distribution of the species contain
the usual random death, birth and migration processes. The evolution
of the mass of the individual particle includes (together with the diffusion)
the mitosis: the splitting of the mass between the two offspring.
The last process leads to the new effects. The limit theorems give the
detailed picture of the space –mass distribution of the particles
in the bulk of the moving front of the population.

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.