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Marek Biskup - One of the best experts on this subject based on the ideXlab platform.

  • a Central Limit Theorem for the effective conductance linear boundary data and small ellipticity contrasts
    Communications in Mathematical Physics, 2014
    Co-Authors: Marek Biskup, Michele Salvi, Tilman Wolff
    Abstract:

    Given a resistor network on \({\mathbb{Z}^d}\) with nearest-neighbor conductances, the effective conductance in a finite set with a given boundary condition is the minimum of the Dirichlet energy over functions with the prescribed boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box converges to a deterministic Limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) Central Limit Theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and ellipticity contrasts will be addressed in a subsequent paper.

  • a Central Limit Theorem for the effective conductance linear boundary data and small ellipticity contrasts
    arXiv: Probability, 2012
    Co-Authors: Marek Biskup, Michele Salvi, Tilman Wolff
    Abstract:

    Given a resistor network on $\mathbb Z^d$ with nearest-neighbor conductances, the effective conductance in a finite set with a given boundary condition is the the minimum of the Dirichlet energy over functions with the prescribed boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box converges to a deterministic Limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) Central Limit Theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and ellipticity contrasts will be addressed in a subsequent paper.

Tilman Wolff - One of the best experts on this subject based on the ideXlab platform.

  • a Central Limit Theorem for the effective conductance linear boundary data and small ellipticity contrasts
    Communications in Mathematical Physics, 2014
    Co-Authors: Marek Biskup, Michele Salvi, Tilman Wolff
    Abstract:

    Given a resistor network on \({\mathbb{Z}^d}\) with nearest-neighbor conductances, the effective conductance in a finite set with a given boundary condition is the minimum of the Dirichlet energy over functions with the prescribed boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box converges to a deterministic Limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) Central Limit Theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and ellipticity contrasts will be addressed in a subsequent paper.

  • a Central Limit Theorem for the effective conductance linear boundary data and small ellipticity contrasts
    arXiv: Probability, 2012
    Co-Authors: Marek Biskup, Michele Salvi, Tilman Wolff
    Abstract:

    Given a resistor network on $\mathbb Z^d$ with nearest-neighbor conductances, the effective conductance in a finite set with a given boundary condition is the the minimum of the Dirichlet energy over functions with the prescribed boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box converges to a deterministic Limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) Central Limit Theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and ellipticity contrasts will be addressed in a subsequent paper.

Cambyse Rouzé - One of the best experts on this subject based on the ideXlab platform.

  • Convergence Rates for the Quantum Central Limit Theorem
    Communications in Mathematical Physics, 2021
    Co-Authors: Simon Becker, Nilanjana Datta, Ludovico Lami, Cambyse Rouzé
    Abstract:

    Various quantum analogues of the Central Limit Theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n -splitter, which is fed with n copies of a centred state $$\rho $$ ρ with finite second moments, converges to the Gaussian state with the same first and second moments as $$\rho $$ ρ . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum Central Limit Theorem. For instance, we prove that the convergence takes place at a rate $$\mathcal {O}\left( n^{-1/2}\right) $$ O n - 1 / 2 in the Hilbert–Schmidt norm whenever the third moments of $$\rho $$ ρ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m -mode state enters a cascade of n beam splitters of equal transmissivities $$\lambda ^{1/n}$$ λ 1 / n fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate $$\mathcal {O}\Big (n^{-\frac{1}{2(m+1)}}\Big )$$ O ( n - 1 2 ( m + 1 ) ) . This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function $$\chi _\rho $$ χ ρ is uniformly bounded by some $$\eta _\rho

  • convergence rates for the quantum Central Limit Theorem
    Communications in Mathematical Physics, 2021
    Co-Authors: Simon Becker, Nilanjana Datta, Ludovico Lami, Cambyse Rouzé
    Abstract:

    Various quantum analogues of the Central Limit Theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state $$\rho $$ with finite second moments, converges to the Gaussian state with the same first and second moments as $$\rho $$ . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum Central Limit Theorem. For instance, we prove that the convergence takes place at a rate $$\mathcal {O}\left( n^{-1/2}\right) $$ in the Hilbert–Schmidt norm whenever the third moments of $$\rho $$ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities $$\lambda ^{1/n}$$ fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate $$\mathcal {O}\Big (n^{-\frac{1}{2(m+1)}}\Big )$$ . This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function $$\chi _\rho $$ is uniformly bounded by some $$\eta _\rho <1$$ outside of any neighbourhood of the origin; also, $$\eta _\rho $$ can be made to depend only on the energy of the state $$\rho $$ .

  • convergence rates for the quantum Central Limit Theorem
    arXiv: Quantum Physics, 2019
    Co-Authors: Simon Becker, Nilanjana Datta, Ludovico Lami, Cambyse Rouzé
    Abstract:

    Various quantum analogues of the Central Limit Theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an $n$-splitter, which is fed with $n$ copies of a centred state $\rho$ with finite second moments, converges to the Gaussian state with the same first and second moments as $\rho$. Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum Central Limit Theorem. For instance, we prove that the convergence takes place at a rate $\mathcal{O}\left(n^{-1/2}\right)$ in the Hilbert--Schmidt norm whenever the third moments of $\rho$ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given single-mode state enters a cascade of $n$ beam splitters of equal transmissivities $\lambda^{1/n}$ fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate $\mathcal{O}\Big(n^{-\frac{1}{2(m+1)}}\Big)$. This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function $\chi_\rho$ is uniformly bounded by some $\eta_\rho<1$ outside of any neighbourhood of the origin; also, $\eta_\rho$ can be made to depend only on the energy of the state $\rho$.

Lingjiong Zhu - One of the best experts on this subject based on the ideXlab platform.

  • Central Limit Theorem for nonlinear hawkes processes
    Journal of Applied Probability, 2013
    Co-Authors: Lingjiong Zhu
    Abstract:

    The Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, finance, and many other fields. In this paper we obtain a functional Central Limit Theorem for the nonlinear Hawkes process. Under the same assumptions, we also obtain a Strassen's invariance principle, i.e. a functional law of the iterated logarithm.

  • Central Limit Theorem for nonlinear hawkes processes
    arXiv: Probability, 2012
    Co-Authors: Lingjiong Zhu
    Abstract:

    Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, finance and many other fields. In this paper, we obtain a functional Central Limit Theorem for nonlinear Hawkes process. Under the same assumptions, we also obtain a Strassen's invariance principle, i.e. a functional law of the iterated logarithm.

Galin L. Jones - One of the best experts on this subject based on the ideXlab platform.

  • On the Markov chain Central Limit Theorem
    Probability Surveys, 2004
    Co-Authors: Galin L. Jones
    Abstract:

    The goal of this expository paper is to describe conditions which guarantee a Central Limit Theorem for functionals of general state space Markov chains. This is done with a view towards Markov chain Monte Carlo settings and hence the focus is on the connections between drift and mixing conditions and their implications. In particular, we consider three commonly cited Central Limit Theorems and discuss their relationship to classical results for mixing processes. Several motivating examples are given which range from toy one-dimensional settings to complicated settings encountered in Markov chain Monte Carlo.