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Zakhar Kabluchko

  1. Extremes of the standardized Gaussian noise.

    Mon, 07/05/2010 - 15:01 — SomNambul
    Authors: Zakhar Kabluchko
    Subjects: Probability
    Abstract

    Let $\{\xi_n, n\in\Z^d\}$ be a $d$-dimensional array of i.i.d. Gaussian
    random variables and define $\SSS(A)=\sum_{n\in A} \xi_n$, where $A$ is a
    finite subset of $\Z^d$. We prove that the appropriately normalized maximum of
    $\SSS(A)/\sqrt{|A|}$, where $A$ ranges over all discrete cubes or rectangles
    contained in $\{1,\ldots,n\}^d$, converges in the weak sense to the Gumbel
    extreme-value distribution as $n\to\infty$. We also prove continuous-time
    counterparts of these results.

  2. Locally adaptive image denoising by a statistical multiresolution criterion.

    Mon, 02/01/2010 - 16:01 — SomNambul
    Authors: Axel Munk, Zakhar Kabluchko, Thomas Hotz, Philipp Marnitz, Rahel Stichtenoth, Laurie Davies
    Subjects: Methodology
    Abstract

    We demonstrate how one can choose the smoothing parameter in image denoising
    by a statistical multiresolution criterion, both globally and locally. Using
    inhomogeneous diffusion and total variation regularization as examples for
    localized regularization schemes, we present an efficient method for locally
    adaptive image denoising. As expected, the smoothing parameter serves as an
    edge detector in this framework. Numerical examples illustrate the usefulness
    of our approach. We also present an application in confocal microscopy.

  3. Functional limit theorems for sums of independent geometric L\'evy processes.

    Tue, 11/24/2009 - 16:01 — SomNambul
    Authors: Zakhar Kabluchko
    Subjects: Probability
    Abstract

    Let $\xi_i$, $i\in\mathbb N$, be independent copies of a L\'evy process
    $\{\xi(t),t\geq 0\}$. Motivated by the results obtained previously in the
    context of the random energy model, we prove functional limit theorems for the
    process $$ Z_N(t)=\sum_{i=1}^N e^{\xi_i(s_N+t)} $$ as $N\to\infty$, where $s_N$
    is a non-negative sequence converging to $+\infty$. The limiting process
    depends heavily on the growth rate of the sequence $s_N$.

  4. Stationary max-stable fields associated to negative definite functions.

    Mon, 09/28/2009 - 15:24 — SomNambul
    Authors: Zakhar Kabluchko, Martin Schlather, Laurens de Haan
    Subjects: Probability
    Abstract

    Let $W_i,i\in{\mathbb{N}}$, be independent copies of a zero-mean Gaussian
    process $\{W(t),t\in{\mathbb{R}}^d\}$ with stationary increments and variance
    $\sigma^2(t)$. Independently of $W_i$, let $\sum_{i=1}^{\infty}\delta_{U_i}$ be
    a Poisson point process on the real line with intensity $e^{-y} dy$. We show
    that the law of the random family of functions
    $\{V_i(\cdot),i\in{\mathbb{N}}\}$, where $V_i(t)=U_i+W_i(t)-\sigma^2(t)/2$, is
    translation invariant. In particular, the process
    $\eta(t)=\bigvee_{i=1}^{\infty}V_i(t)$ is a stationary max-stable process with
    standard Gumbel margins.

  5. Stationary max-stable fields associated to negative definite functions.

    Mon, 09/28/2009 - 15:24 — SomNambul
    Authors: Zakhar Kabluchko, Martin Schlather, Laurens de Haan
    Subjects: Probability
    Abstract

    Let $W_i,i\in{\mathbb{N}}$, be independent copies of a zero-mean Gaussian
    process $\{W(t),t\in{\mathbb{R}}^d\}$ with stationary increments and variance
    $\sigma^2(t)$. Independently of $W_i$, let $\sum_{i=1}^{\infty}\delta_{U_i}$ be
    a Poisson point process on the real line with intensity $e^{-y} dy$. We show
    that the law of the random family of functions
    $\{V_i(\cdot),i\in{\mathbb{N}}\}$, where $V_i(t)=U_i+W_i(t)-\sigma^2(t)/2$, is
    translation invariant. In particular, the process
    $\eta(t)=\bigvee_{i=1}^{\infty}V_i(t)$ is a stationary max-stable process with
    standard Gumbel margins.

  6. Extremes of Independent Gaussian Processes.

    Thu, 09/03/2009 - 17:38 — SomNambul
    Authors: Zakhar Kabluchko
    Subjects: Probability
    Abstract

    For every $n\in\N$, let $X_{1n},..., X_{nn}$ be independent copies of a
    zero-mean Gaussian process $X_n=\{X_n(t), t\in T\}$. We describe all processes
    which can be obtained as limits, as $n\to\infty$, of the process
    $a_n(M_n-b_n)$, where $M_n(t)=\max_{i=1,...,n} X_{in}(t)$ and $a_n, b_n$ are
    normalizing constants. We also provide an analogous characterization for the
    limits of the process $a_nL_n$, where $L_n(t)=\min_{i=1,...,n} |X_{in}(t)|$.

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