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Thomas Mikosch

  1. Estimating Extremal Dependence in Univariate and Multivariate Time Series via the Extremogram.

    Fri, 07/29/2011 - 15:01 — SomNambul
    Authors: Thomas Mikosch, Richard A. Davis, Ivor Cribben
    Subjects: Methodology
    Abstract

    Davis and Mikosch [7] introduced the extremogram as a flexible quantitative
    tool for measuring various types of extremal dependence in a stationary time
    series. There we showed some standard statistical properties of the sample
    extremogram. A major difficulty was the construction of credible confidence
    bands for the extremogram. In this paper, we employ the stationary bootstrap to
    overcome this problem. Moreover, we introduce the cross extremogram as a
    measure of extremal serial dependence between two or more time series.

  2. The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution.

    Mon, 11/29/2010 - 16:01 — SomNambul
    Authors: Thomas Mikosch, Alfredas Račkauskas
    Subjects: Statistics
    Abstract

    In this paper, we deal with the asymptotic distribution of the maximum
    increment of a random walk with a regularly varying jump size distribution.
    This problem is motivated by a long-standing problem on change point detection
    for epidemic alternatives. It turns out that the limit distribution of the
    maximum increment of the random walk is one of the classical extreme value
    distributions, the Fr\'{e}chet distribution. We prove the results in the
    general framework of point processes and for jump sizes taking values in a
    separable Banach space.

  3. Weak convergence of the function-indexed integrated periodogram for infinite variance processes.

    Wed, 11/24/2010 - 16:01 — SomNambul
    Authors: Gennady Samorodnitsky, Thomas Mikosch, Sami Umut Can
    Subjects: Statistics
    Abstract

    In this paper, we study the weak convergence of the integrated periodogram
    indexed by classes of functions for linear processes with symmetric
    $\alpha$-stable innovations. Under suitable summability conditions on the
    series of the Fourier coefficients of the index functions, we show that the
    weak limits constitute $\alpha$-stable processes which have representations as
    infinite Fourier series with i.i.d. $\alpha$-stable coefficients. The cases
    $\alpha\in(0,1)$ and $\alpha\in[1,2)$ are dealt with by rather different
    methods and under different assumptions on the classes of functions.

  4. The extremogram: A correlogram for extreme events.

    Wed, 01/13/2010 - 16:01 — SomNambul
    Authors: Thomas Mikosch, Richard A. Davis
    Subjects: Statistics
    Abstract

    We consider a strictly stationary sequence of random vectors whose
    finite-dimensional distributions are jointly regularly varying with some
    positive index. This class of processes includes, among others, ARMA processes
    with regularly varying noise, GARCH processes with normally or
    Student-distributed noise and stochastic volatility models with regularly
    varying multiplicative noise. We define an analog of the autocorrelation
    function, the extremogram, which depends only on the extreme values in the
    sequence.

  5. Infinite variance stable limits for sums of dependent random variables.

    Tue, 09/08/2009 - 12:16 — SomNambul
    Authors: Katarzyna Bartkiewicz, Adam Jakubowski, Thomas Mikosch, Olivier Wintenberger
    Subjects: Probability
    Abstract

    The aim of this paper is to provide conditions which ensure that the affinely
    transformed partial sums of a strictly stationary process converge in
    distribution to an in?nite variance stable distribution. Conditions for this
    convergence to hold are known in the literature. However, most of these results
    are qualitative in the sense that the parameters of the limit distribution are
    expressed in terms of some limiting point process.

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