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School of Mathematics

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search.filters.applied.f.advisor: Houdré, Christian × End date: 2009 × Start date: 2000 ×

Publication Search Results

Now showing 1 - 6 of 6
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    On the limiting shape of random young tableaux for Markovian words
    (Georgia Institute of Technology, 2008-11-17) Litherland, Trevis J.
    The limiting law of the length of the longest increasing subsequence, LI_n, for sequences (words) of length n arising from iid letters drawn from finite, ordered alphabets is studied using a straightforward Brownian functional approach. Building on the insights gained in both the uniform and non-uniform iid cases, this approach is then applied to iid countable alphabets. Some partial results associated with the extension to independent, growing alphabets are also given. Returning again to the finite setting, and keeping with the same Brownian formalism, a generalization is then made to words arising from irreducible, aperiodic, time-homogeneous Markov chains on a finite, ordered alphabet. At the same time, the probabilistic object, LI_n, is simultaneously generalized to the shape of the associated Young tableau given by the well-known RSK-correspondence. Our results on this limiting shape describe, in detail, precisely when the limiting shape of the Young tableau is (up to scaling) that of the iid case, thereby answering a conjecture of Kuperberg. These results are based heavily on an analysis of the covariance structure of an m-dimensional Brownian motion and the precise form of the Brownian functionals. Finally, in both the iid and more general Markovian cases, connections to the limiting laws of the spectrum of certain random matrices associated with the Gaussian Unitary Ensemble (GUE) are explored.
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    Aspects of random matrix theory: concentration and subsequence problems
    (Georgia Institute of Technology, 2008-11-17) Xu, Hua
    The present work studies some aspects of random matrix theory. Its first part is devoted to the asymptotics of random matrices with infinitely divisible, in particular heavy-tailed, entries. Its second part focuses on relations between limiting law in subsequence problems and spectra of random matrices.
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    Nonparametric estimation of Levy processes with a view towards mathematical finance
    (Georgia Institute of Technology, 2004-04-08) Figueroa-Lopez, Jose Enrique
    Model selection methods and nonparametric estimation of Levy densities are presented. The estimation relies on the properties of Levy processes for small time spans, on the nature of the jumps of the process, and on methods of estimation for spatial Poisson processes. Given a linear space S of possible Levy densities, an asymptotically unbiased estimator for the orthogonal projection of the Levy density onto S is found. It is proved that the expected standard error of the proposed estimator realizes the smallest possible distance between the true Levy density and the linear space S as the frequency of the data increases and as the sampling time period gets longer. Also, we develop data-driven methods to select a model among a collection of models. The method is designed to approximately realize the best trade-off between the error of estimation within the model and the distance between the model and the unknown Levy density. As a result of this approach and of concentration inequalities for Poisson functionals, we obtain Oracles inequalities that guarantee us to reach the best expected error (using projection estimators) up to a constant. Numerical results are presented for the case of histogram estimators and variance Gamma processes. To calibrate parametric models,a nonparametric estimation method with least-squares errors is studied. Comparison with maximum likelihood estimation is provided. On a separate problem, we review the theoretical properties of temepered stable processes, a class of processes with potential great use in Mathematical Finance.
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    Curvature, isoperimetry, and discrete spin systems
    (Georgia Institute of Technology, 2001-12) Murali, Shobhana
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    Isoperimetic and related constants for graphs and markov chains
    (Georgia Institute of Technology, 2001-08) Stoyanov, Tsvetan I.
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    Dependence and limit theorems for stationary infinitely divisible sequences
    (Georgia Institute of Technology, 2000-08) Harrelson, Dyana Rae