The Experts below are selected from a list of 258 Experts worldwide ranked by ideXlab platform

Vladimir V. V'yugin - One of the best experts on this subject based on the ideXlab platform.

  • On Stability Property of Probability Laws with Respect to Small Violations of Algorithmic Randomness
    arXiv: Computational Complexity, 2014
    Co-Authors: Vladimir V. V'yugin
    Abstract:

    We study a stability property of probability laws with respect to small violations of algorithmic randomness. A sufficient condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like the strong law of large numbers, the law of iterated logarithm, and even Birkhoff's pointwise ergodic theorem for ergodic Transformations, are stable in this sense. Nevertheless, the phenomenon of instability occurs in ergodic theory. Firstly, the stability property of the Birkhoff's ergodic theorem is non-uniform. Moreover, a computable non-ergodic Measure Preserving Transformation can be constructed such that ergodic theorem is non-stable. We also show that any universal data compression scheme is also non-stable with respect to the class of all computable ergodic Measures.

V. V. Ryzhikov - One of the best experts on this subject based on the ideXlab platform.

Mikhail Gordin - One of the best experts on this subject based on the ideXlab platform.

  • Limit theorems for von Mises statistics of a Measure Preserving Transformation
    Probability Theory and Related Fields, 2014
    Co-Authors: Manfred Denker, Mikhail Gordin
    Abstract:

    For a Measure Preserving Transformation $$T$$ T of a probability space $$(X,\mathcal{F },\mu )$$ ( X , F , μ ) and some $$d \ge 1$$ d ≥ 1 we investigate almost sure and distributional convergence of random variables of the form $$\begin{aligned} x \rightarrow \frac{1}{C_n} \sum _{0\le i_1,\ldots ,\,i_d

  • limit theorems for von mises statistics of a Measure Preserving Transformation
    Probability Theory and Related Fields, 2014
    Co-Authors: Manfred Denker, Mikhail Gordin
    Abstract:

    For a Measure Preserving Transformation \(T\) of a probability space \((X,\mathcal{F },\mu )\) and some \(d \ge 1\) we investigate almost sure and distributional convergence of random variables of the form $$\begin{aligned} x \rightarrow \frac{1}{C_n} \sum _{0\le i_1,\ldots ,\,i_dsubspace in some \(L_p(X^d\!,\, \mathcal{F }^{\otimes d}\!,\,\mu ^d)\). We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with \(T\) and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and \(d=2\), we also show that the convergence holds in distribution towards a quadratic form \(\sum _{m=1}^{\infty } \lambda _m\eta ^2_m\) in independent standard Gaussian variables \(\eta _1, \eta _2, \ldots \).

  • limit theorems for von mises statistics of a Measure Preserving Transformation
    arXiv: Dynamical Systems, 2011
    Co-Authors: Manfred Denker, Mikhail Gordin
    Abstract:

    For a Measure Preserving Transformation $T$ of a probability space $(X,\mathcal F,\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \to \frac{1}{C_n} \sum_{i_1random variables are well defined and belong to $L_r(\mu)$ provided that the kernel is chosen from the projective tensor product $$L_p(X_1,\mathcal F_1, \mu_1) \otimes_{\pi}...\otimes_{\pi} L_p(X_d,\mathcal F_d, \mu_d)\subset L_p(\mu^d)$$ with $p=d\,r,\, r\ \in [1, \infty).$ We establish a form of the individual ergodic theorem for such sequences. Next, we give a martingale approximation argument to derive a central limit theorem in the non-degenerate case (in the sense of the classical Hoeffding's decomposition). Furthermore, for $d=2$ and a wide class of canonical kernels $f$ we also show that the convergence holds in distribution towards a quadratic form $\sum_{m=1}^{\infty} \lambda_m\eta^2_m$ in independent standard Gaussian variables $\eta_1, \eta_2,...$. Our results on the distributional convergence use a $T$--\,invariant filtration as a prerequisite and are derived from uni- and multivariate martingale approximations.

Abbas Moameni - One of the best experts on this subject based on the ideXlab platform.

  • symmetric monge kantorovich problems and polar decompositions of vector fields
    Geometric and Functional Analysis, 2014
    Co-Authors: Nassif Ghoussoub, Abbas Moameni
    Abstract:

    We address the problem of whether a bounded measurable vector field from a bounded domain Ω into \({\mathbb{R}^d}\) is N-cyclically monotone up to a Measure Preserving N-involution, where N is any integer larger than 2. Our approach involves the solution of a multidimensional symmetric Monge–Kantorovich problem, which we first study in the case of a general cost function on a product domain ΩN. The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually N − 1 of them). The problem amounts to showing that the supremum in the corresponding Monge–Kantorovich problem when restricted to those probability Measures on ΩN which are invariant under cyclic permutations and with a given first marginal μ, is attained on a probability Measure that is supported on a graph of the form x → (x, Sx, S2x,..., SN-1x), where S is a μ-Measure Preserving Transformation on Ω such that SN = I a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are N-cyclically antisymmetric.

  • symmetric monge kantorovich problems and polar decompositions of vector fields
    arXiv: Analysis of PDEs, 2013
    Co-Authors: Nassif Ghoussoub, Abbas Moameni
    Abstract:

    For any given integer $N\geq 2$, we show that every bounded measurable vector field from a bounded domain $\Omega$ into $\R^d$ is $N$-cyclically monotone up to a Measure Preserving $N$-involution. The proof involves the solution of a multidimensional symmetric Monge-Kantorovich problem, which we first study in the case of a general cost function on a product domain $\Omega^N$. The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually $N-1$ of them). In this case, we show that the supremum over all probability Measures on $\Omega^N$ which are invariant under cyclic permutations and with a given first marginal $\mu$, is attained on a probability Measure that is supported on the graph of a function of the form $x\to (x, Sx, S^2x,..., S^{N-1}x)$, where $S$ is a $\mu$-Measure Preserving Transformation on $\Omega$ such that $S^N=I$ a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are $N$-cyclically antisymmetric.

Alexandre I. Danilenko - One of the best experts on this subject based on the ideXlab platform.