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Faculties » Faculty of Mathematics » Chairs » Lehrstuhl XII: Stochastik

Research

New Techniques for Empirical Processes of Dependent Data (DFG Grant DE 370-4)

In this cooperational project with Dr. Olivier Durieu of Université François-Rabelais de Tours (France), we investigate the asymptotic behavior of empirical processes of multidimensional dependent data. In an initial work, Dehling, Durieu, and Volny (2009) developed a new technique to prove empirical central limit theorems for real-valued stationary dependent data. Dehling et al. (2009) were mainly interested in stationary processes, where the central limit theorem can be established under functionals from certain class of functions using the spectral gap technique. In a later work, Dehling and Durieu (2011) expanded their technique to applications on processes that possess a multiple mixing property. In this context they were e.g. able to prove an empirical CLT for ergodic automorphisms of the multidimensional torus.

In the situations mentioned above Prokhorov's classical “finite-dimensional convergence plus tightness” approach seems quite difficult to apply, since one does not have the finite dimensional convergence to start with. Instead, Dehling et al. (2009) developed a technique based on finite-dimensional approximations of the empirical process by empirical process indexed by a different class of functions that feature better analytic properties than the functions from the original class. Eventually making use of a modification Theorem of Billingsley (1968, Theorem 4.2), Dehling, Durieu, and Volný (2009) were able to determine the asymptotic behaviour of the original process by that of the approximating process.

During my research, we were able to extend this technique to the following situations.

• processes of a weaker mixing property: We show how to establish empirical CLT's for processes with a slow (e.g. polynomial) decay of dependence.
• empirical processes, indexed by classes of functions: We consider empirical processes that are indexed by more general classes of functions than the usual indicatior fuctions of semifinite rectangles.
• sequential empirical processes: Recently, we extended our technique to sequential empirical processes, i.e. processes where the indexing set is extended by a time parameter limiting the number of summands in the partial sum process. Sequential empirical process CLTs (i.e. the convergence of a sequential empirical process to a so called Kiefer-Müller process) find application in the study of the asymptotic distribution of change-point tests based on the empirical distribution function.

Current Research

At the moment I am interested in the study of so called weighted empirical processes, that is an empirical process multiplied with some weight function. The convergence of such processes can be used among others to deduce the asymptotic behaviour of empirical U-process. Currently we are developing a technique to treat weighted empirical processes of underlying multiple mixing data.

Publications

H. Dehling, O. Durieu, and M. Tusche (2014 b): A Sequential Empirical Central Limit Theorem for Multiple Mixing Processes with Application to B-Geometrically Ergodic Markov Chains. Electronic Journal of Probability, 19, article 87.   [arXiv 1303.4537].

H. Dehling, O. Durieu, and M. Tusche (2014 a): Approximating Class Approach for Empirical Processes of Dependent Sequences Indexed by Functions. Bernoulli. 20, no. 3, p. 1372–1403.   [arXiv 1201.2256].

M. Tusche (2014): Empirical Processes of Multiple Mixing Data. Dissertation. Ruhr-University Bochum and Université François-Rabelais de Tours.

A. El-Hadidy, C. Rehtanz, M. Tusche, and C. Langesberg (2013): Investigation of the effect of current and voltage transformers' errors on a differential-like impedance relaying algorithm — Protection from a probabilistic view . PowerTech, IEEE Grenoble. Published electronically 2013. DOI: 10.1109/PTC.2013.6652261. Online available on IEEE Xplore Digital Library.

O. Durieu, M. Tusche (2012): An empirical process central limit theorem for multidimensional dependent data. Journal of Theoretical Probability. 27, no. 1, p. 249-277.   [arXiv 1110.0963].

Poster: An Outer Bracketing Approach for Empirical Process CLTs of Weakly Dependent Data

References

P. Billingsley (1968): Convergence of Probability Measures. John Wiley & Sons Inc., New York.

H. Dehling, O. Durieu, and D. Volný (2009): New techniques for empirical processes of dependent data. Stochastic Process. Appl. 119, no. 10, p. 3699–3718

H. Dehling, O. Durieu (2011): Empirical processes of multidimensional systems with multiple mixing properties. Stochastic Process. Appl. 121, no. 5, p. 1076–1096

Miscellaneous

Interdisciplinary Cooperation

In a chat with a friend about our fields of science, we came up with the idea to treat a certain problem from the field of electrical engineering with the cumulative distribution function. This approach was developed futher and was presented at a conference of the "Institute of Electrical and Electronics Engineers" (IEEE) in Grenoble in 2013. It led to the conference paper El-Hadidy, Rehtanz, Tusche, and Langesberg (2013), see above.

Doctoral Student Conference in Stochastics 2011

Together with Katharina Proksch, I organized the seventh Doctoral Student Conference in Stochastics ("Doktorandentreffen Stochastik"), which took place in autumn 2011 at Ruhr-University Bochum. The conference was addressed to doctoral students of stochastics from Austrian, German, and Swiss universities. The attendants had the opportunity to outline their fields of research and results as well as to gain an overview of the area of work of other postgraduate students.

Diploma Thesis

In change-point analysis the maximum increment of a random walk can be a useful test statistic. Mikosch and Račkauskas (2010) showed that the distribution of a random walk with underlying independent identically distributed heavy-tailed random variables converges under moderate conditions weakly to a Frechet distribution (extreme value distribution).
In my diploma thesis (german), a full detailed discussion of Mikosch's and Račkauskas' proof is given. Especially the necessary theory of random measures and point processes (including all relevant topological and probability theoretical foundations) is introduced and illustrated in its applications.

T. Mikosch and Alfredas Račkauskas (2010): The limit distribution of the maximum increment of a heavy-tailed random walk. Bernoulli 16, no.4, 1016-1038.   Preprint: [State: 09.04.2009]

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