231 research outputs found

    On a model selection problem from high-dimensional sample covariance matrices

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    Modern random matrix theory indicates that when the population size p is not negligible with respect to the sample size n, the sample covariance matrices demonstrate significant deviations from the population covariance matrices. In order to recover the characteristics of the population covariance matrices from the observed sample covariance matrices, several recent solutions are proposed when the order of the underlying population spectral distribution is known. In this paper, we deal with the underlying order selection problem and propose a solution based on the cross-validation principle. We prove the consistency of the proposed procedure. © 2011 Elsevier Inc.postprin

    Lp solutions of backward stochastic differential equations

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    AbstractIn this paper, we are interested in solving backward stochastic differential equations (BSDEs for short) under weak assumptions on the data. The first part of the paper is devoted to the development of some new technical aspects of stochastic calculus related to BSDEs. Then we derive a priori estimates and prove existence and uniqueness of solutions in Lp p>1, extending the results of El Karoui et al. (Math. Finance 7(1) (1997) 1) to the case where the monotonicity conditions of Pardoux (Nonlinear Analysis; Differential Equations and Control (Montreal, QC, 1998), Kluwer Academic Publishers, Dordrecht, pp. 503–549) are satisfied. We consider both a fixed and a random time interval. In the last section, we obtain, under an additional assumption, an existence and uniqueness result for BSDEs on a fixed time interval, when the data are only in L1

    Metal-insulator transition in the Hartree-Fock phase diagram of the fully polarized homogeneous electron gas in two dimensions

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    We determine numerically the ground state of the two-dimensional, fully polarized electron gas within the Hartree-Fock approximation without imposing any particular symmetries on the solutions. At low electronic densities, the Wigner crystal solution is stable, but for higher densities (rsr_s less than ∼2.7\sim 2.7) we obtain a ground state of different symmetry: the charge density forms a triangular lattice with about 11% more sites than electrons. We prove analytically that this conducting state with broken translational symmetry has lower energy than the uniform Fermi gas state in the high density region giving rise to a metal to insulator transition.Comment: 13 pages, 5 figures, rewrite of 0804.1025 and 0807.077

    Localization and Mobility Edge in One-Dimensional Potentials with Correlated Disorder

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    We show that a mobility edge exists in 1D random potentials provided specific long-range correlations. Our approach is based on the relation between binary correlator of a site potential and the localization length. We give the algorithm to construct numerically potentials with mobility edge at any given energy inside allowed zone. Another natural way to generate such potentials is to use chaotic trajectories of non-linear maps. Our numerical calculations for few particular potentials demonstrate the presence of mobility edges in 1D geometry.Comment: 4 pages in RevTex and 2 Postscript figures; revised version published in Phys. Rev. Lett. 82 (1999) 406

    Diffusion in disordered systems under iterative measurement

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    We consider a sequence of idealized measurements of time-separation Δt\Delta t onto a discrete one-dimensional disordered system. A connection with Markov chains is found. For a rapid sequence of measurements, a diffusive regime occurs and the diffusion coefficient DD is analytically calculated. In a general point of view, this result suggests the possibility to break the Anderson localization due to decoherence effects. Quantum Zeno effect emerges because the diffusion coefficient DD vanishes at the limit Δt→0\Delta t \to 0.Comment: 8 pages, 0 figures, LATEX. accepted in Phys.Rev.

    The scaling limit of the critical one-dimensional random Schrodinger operator

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    We consider two models of one-dimensional discrete random Schrodinger operators (H_n \psi)_l ={\psi}_{l-1}+{\psi}_{l +1}+v_l {\psi}_l, {\psi}_0={\psi}_{n+1}=0 in the cases v_k=\sigma {\omega}_k/\sqrt{n} and v_k=\sigma {\omega}_k/ \sqrt{k}. Here {\omega}_k are independent random variables with mean 0 and variance 1. We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem. In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the beta-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.Comment: 36 pages, 2 figure

    A stochastic approximation algorithm with multiplicative step size modification

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    An algorithm of searching a zero of an unknown function \vphi : \, \R \to \R is considered:  xt=xt−1−γt−1yt\, x_{t} = x_{t-1} - \gamma_{t-1} y_t,\, t=1, 2,…t=1,\ 2,\ldots, where yt=φ(xt−1)+ξty_t = \varphi(x_{t-1}) + \xi_t is the value of \vphi measured at xt−1x_{t-1} and ξt\xi_t is the measurement error. The step sizes \gam_t > 0 are modified in the course of the algorithm according to the rule: \, \gamma_t = \min\{u\, \gamma_{t-1},\, \mstep\} if yt−1yt>0y_{t-1} y_t > 0, and γt=d γt−1\gamma_t = d\, \gamma_{t-1}, otherwise, where 0<d<100 < d < 1 0. That is, at each iteration \gam_t is multiplied either by uu or by dd, provided that the resulting value does not exceed the predetermined value \mstep. The function \vphi may have one or several zeros; the random values ξt\xi_t are independent and identically distributed, with zero mean and finite variance. Under some additional assumptions on \vphi, ξt\xi_t, and \mstep, the conditions on uu and dd guaranteeing a.s. convergence of the sequence {xt}\{ x_t \}, as well as a.s. divergence, are determined. In particular, if ¶(ξ1>0)=¶(ξ1<0)=1/2\P (\xi_1 > 0) = \P (\xi_1 < 0) = 1/2 and ¶(ξ1=x)=0\P (\xi_1 = x) = 0 for any x∈Rx \in \R, one has convergence for ud1ud 1. Due to the multiplicative updating rule for \gam_t, the sequence {xt}\{ x_t \} converges rapidly: like a geometric progression (if convergence takes place), but the limit value may not coincide with, but instead, approximates one of the zeros of \vphi. By adjusting the parameters uu and dd, one can reach arbitrarily high precision of the approximation; higher precision is obtained at the expense of lower convergence rate
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