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Mu Sun - One of the best experts on this subject based on the ideXlab platform.
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noncommutative multi parameter wiener wintner type Ergodic Theorem
Journal of Functional Analysis, 2018Co-Authors: Guixiang Hong, Mu SunAbstract:Abstract In this paper, we establish a multi-parameter version of Bellow and Losert's Wiener–Wintner type Ergodic Theorem for dynamical systems not necessarily commutative. More precisely, we introduce a weight class D , which is shown to strictly include the multi-parameter bounded Besicovitch weight class, thus including the set Λ d = { { λ 1 k 1 ⋯ λ d k d } ( k 1 , … , k d ) ∈ N d : ( λ 1 , … , λ d ) ∈ T d } ; then we prove a multi-parameter Bellow and Losert's Wiener–Wintner type Ergodic Theorem for the class D and for a noncommutative trace preserving dynamical system ( M , τ , T ) , M being a von Neumann algebra. Restricted to Λ d , we also prove a noncommutative multi-parameter analogue of Bourgain's uniform Wiener–Wintner Ergodic Theorem. The “noncommutativity” and the “multi-parameter” characters induce some difficulties in the proofs. For instance, our argument of proving the uniform convergence for a dense subset turns out to be quite different from the classical case since the “pointwise” argument does not work in the noncommutative setting; also to obtain the uniform convergence in the largest spaces, we need a maximal inequality between the Orlicz spaces, but it cannot be deduced by using classical extrapolation argument directly. Junge and Xu's noncommutative maximal inequalities with the optimal order, together with the atomic decomposition of Orlicz spaces, play the essential role in overcoming the second difficulty.
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noncommutative multi parameter wiener wintner type Ergodic Theorem
arXiv: Operator Algebras, 2016Co-Authors: Guixiang Hong, Mu SunAbstract:In this paper, we establish a multi-parameter version of Bellow and Losert's Wiener-Wintner type Ergodic Theorem for dynamical systems not necessarily being commutative. More precisely, we introduce a weight class $\mathcal{D}$, which is shown to strictly include the multi-parameter bounded Besicovitch weight class, thus including the set $$\Lambda_d=\left\{\{\lambda^{k_1}_1\dotsm\lambda^{k_d}_d\}_{(k_1,\dots,k_d)\in \mathbb{N}^d}:\quad (\lambda_1,\dots,\lambda_d) \in \mathbb{T}^d\right\};$$ then prove a multi-parameter Bellow and Losert's Wiener-Wintner type Ergodic Theorem for the class $\mathcal{D}$ and for noncommutative trace preserving dynamical system $(\mathcal{M},\tau,\mathbf{T})$. Restricted to consider the set $\Lambda_d$, we also prove a noncommutative multi-parameter analogue of Bourgain's uniform Wiener-Wintner Ergodic Theorem. The noncommutativity and the multi-parameter induce some difficulties in the proof. For instance, our arguments in proving the uniform convergence for a dense subset turn out to be quite different since the "pointwise" argument does not work in the noncommutative setting; to obtain the uniform convergence in the largest spaces, we show maximal inequality between the Orlicz spaces, which can not be deduced easily using classical extrapolation argument. Junge and Xu's noncommutative maximal inequalities with optimal order, together with the atomic decomposition of Orlicz spaces, play an essential role in overcoming the second difficulty.
Guixiang Hong - One of the best experts on this subject based on the ideXlab platform.
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noncommutative multi parameter wiener wintner type Ergodic Theorem
Journal of Functional Analysis, 2018Co-Authors: Guixiang Hong, Mu SunAbstract:Abstract In this paper, we establish a multi-parameter version of Bellow and Losert's Wiener–Wintner type Ergodic Theorem for dynamical systems not necessarily commutative. More precisely, we introduce a weight class D , which is shown to strictly include the multi-parameter bounded Besicovitch weight class, thus including the set Λ d = { { λ 1 k 1 ⋯ λ d k d } ( k 1 , … , k d ) ∈ N d : ( λ 1 , … , λ d ) ∈ T d } ; then we prove a multi-parameter Bellow and Losert's Wiener–Wintner type Ergodic Theorem for the class D and for a noncommutative trace preserving dynamical system ( M , τ , T ) , M being a von Neumann algebra. Restricted to Λ d , we also prove a noncommutative multi-parameter analogue of Bourgain's uniform Wiener–Wintner Ergodic Theorem. The “noncommutativity” and the “multi-parameter” characters induce some difficulties in the proofs. For instance, our argument of proving the uniform convergence for a dense subset turns out to be quite different from the classical case since the “pointwise” argument does not work in the noncommutative setting; also to obtain the uniform convergence in the largest spaces, we need a maximal inequality between the Orlicz spaces, but it cannot be deduced by using classical extrapolation argument directly. Junge and Xu's noncommutative maximal inequalities with the optimal order, together with the atomic decomposition of Orlicz spaces, play the essential role in overcoming the second difficulty.
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noncommutative multi parameter wiener wintner type Ergodic Theorem
arXiv: Operator Algebras, 2016Co-Authors: Guixiang Hong, Mu SunAbstract:In this paper, we establish a multi-parameter version of Bellow and Losert's Wiener-Wintner type Ergodic Theorem for dynamical systems not necessarily being commutative. More precisely, we introduce a weight class $\mathcal{D}$, which is shown to strictly include the multi-parameter bounded Besicovitch weight class, thus including the set $$\Lambda_d=\left\{\{\lambda^{k_1}_1\dotsm\lambda^{k_d}_d\}_{(k_1,\dots,k_d)\in \mathbb{N}^d}:\quad (\lambda_1,\dots,\lambda_d) \in \mathbb{T}^d\right\};$$ then prove a multi-parameter Bellow and Losert's Wiener-Wintner type Ergodic Theorem for the class $\mathcal{D}$ and for noncommutative trace preserving dynamical system $(\mathcal{M},\tau,\mathbf{T})$. Restricted to consider the set $\Lambda_d$, we also prove a noncommutative multi-parameter analogue of Bourgain's uniform Wiener-Wintner Ergodic Theorem. The noncommutativity and the multi-parameter induce some difficulties in the proof. For instance, our arguments in proving the uniform convergence for a dense subset turn out to be quite different since the "pointwise" argument does not work in the noncommutative setting; to obtain the uniform convergence in the largest spaces, we show maximal inequality between the Orlicz spaces, which can not be deduced easily using classical extrapolation argument. Junge and Xu's noncommutative maximal inequalities with optimal order, together with the atomic decomposition of Orlicz spaces, play an essential role in overcoming the second difficulty.
Weiguo Yang - One of the best experts on this subject based on the ideXlab platform.
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markov approximation and the generalized entropy Ergodic Theorem for non null stationary process
Proceedings - Mathematical Sciences, 2020Co-Authors: Zhongzhi Wang, Weiguo YangAbstract:In an earlier work, we proved a generalized entropy Ergodic Theorem for finite nonhomogeneous Markov chains (NMC). In this paper, we establish a generalized strong law of large numbers for finite m-th order NMC. Then we deduce a generalized entropy Ergodic Theorem for finite m-th order NMC, under some assumptions on the continuity rate and of non-nullness. Explicit upper and lower bounds relating the generalized relative entropy density of the original finite non-null stationary sequence and its canonical m-order Markov approximation is obtained.
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the generalized entropy Ergodic Theorem for nonhomogeneous markov chains
Journal of Theoretical Probability, 2016Co-Authors: Zhongzhi Wang, Weiguo YangAbstract:Let \((\xi _n)_{n=0}^\infty \) be a nonhomogeneous Markov chain taking values in a finite state-space \(\mathbf {X}=\{1,2,\ldots ,b\}\). In this paper, we will study the generalized entropy Ergodic Theorem with almost-everywhere and \(\mathcal {L}_1\) convergence for nonhomogeneous Markov chains; this generalizes the corresponding classical results for Markov chains.
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the strong law of large numbers and the entropy Ergodic Theorem for nonhomogeneous bifurcating markov chains indexed by a binary tree
IEEE Transactions on Information Theory, 2015Co-Authors: Hui Dang, Weiguo Yang, Zhiyan ShiAbstract:Guyon (Guyon J. Limit Theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann Appl Probab, 2007, 17: 1538-1569) introduced an important model for homogeneous bifurcating Markov chains indexed by a binary tree taking values in general state space and studied their limit Theorems. The results were applied to detect cellular aging. In this paper, we define a discrete form of nonhomogeneous bifurcating Markov chains indexed by a binary tree and discuss the equivalent properties for them. The strong law of large numbers and the entropy Ergodic Theorem are studied for these Markov chains with finite state space. In contrast to previous work, we use a new approach to prove the main results of this paper.
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the generalized entropy Ergodic Theorem for nonhomogeneous markov chains
arXiv: Probability, 2015Co-Authors: Zhongzhi Wang, Weiguo YangAbstract:Let $(\xi_n)_{n=0}^\infty$ be a nonhomogeneous Markov chain taking values from finite state-space of $\mathbf{X}=\{1,2,\ldots,b\}$. In this paper, we will study the generalized entropy Ergodic Theorem with almost-everywhere and $\mathcal{L}_1$ convergence for nonhomogeneous Markov chains, which generalizes the corresponding classical results for the Markov chains.
G A Margulis - One of the best experts on this subject based on the ideXlab platform.
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a multiplicative Ergodic Theorem and nonpositively curved spaces
Communications in Mathematical Physics, 1999Co-Authors: Anders Karlsson, G A MargulisAbstract:We study integrable cocycles u.n; x/ over an Ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y , e.g. a Cartan-Hadamard space or a uniformly convex Banach space. It is proved that for any y 2 Y and almost all x; there exist A 0 and a unique geodesic ray .t; x/in Y starting at y such that lim n!1 1 n d. .An; x/; u.n; x/y/ D 0: In the case where Y is the symmetric space GLN.R/=ON.R/ and the cocycles take values in GLN.R/; this is equivalent to the multiplicative Ergodic Theorem of Oseledec. Two applications are also described. The first concerns the determination of Poisson boundaries and the second concerns Hilbert-Schmidt operators.
Semyon Litvinov - One of the best experts on this subject based on the ideXlab platform.
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the validity space of dunford schwartz pointwise Ergodic Theorem
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Vladimir Chilin, Semyon LitvinovAbstract:Abstract We show that if a σ-finite infinite measure space ( Ω , μ ) is quasi-non-atomic, then the Dunford–Schwartz pointwise Ergodic Theorem holds for f ∈ L 1 ( Ω ) + L ∞ ( Ω ) if and only if μ { | f | ≥ λ } ∞ for all λ > 0 .
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validity space of dunford schwartz pointwise Ergodic Theorem
arXiv: Functional Analysis, 2017Co-Authors: Vladimir Chilin, Semyon LitvinovAbstract:We show that if a $\sigma-$finite infinite measure space $(\Omega,\mu)$ is quasi-non-atomic, then the Dunford-Schwartz pointwise Ergodic Theorem holds for $f\in \mathcal L^1(\Omega)+\mathcal L^{\infty}(\Omega)$ if and only if $\mu\{f\ge \lambda\} 0$.
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the validity space of dunford schwartz Ergodic Theorem for infinite measure
arXiv: Functional Analysis, 2017Co-Authors: Vladimir Chilin, Semyon LitvinovAbstract:We show that if $(\Omega,\mu)$ is an infinite measure space, the pointwise Dunford-Shwartz Ergodic Theorem holds for $f \in \mathcal L^1(\Omega)+\mathcal L^\infty(\Omega)$ if and only if $\mu\{f>\lambda\} 0$.
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on individual subsequential Ergodic Theorem in von neumann algebras
Studia Mathematica, 2001Co-Authors: Semyon Litvinov, Farrukh MukhamedovAbstract:We use a non-commutative generalization of the Banach Principle to show that the classical individual Ergodic Theorem for subsequences generated by means of uniform sequences can be extended to the von Neumann algebra setting. 0. Introduction. The study of almost everywhere convergence of the Ergodic averages in the non-commutative setting was initiated by a number of authors among whom we mention Lance (5) and Yeadon (11). Individual Ergodic Theorems have been established for algebras with states as well as for algebras equipped with a seminite trace. The study of almost everywhere convergence of weighted and subsequential averages in von Neumann alge- bras is relatively new. So far, not much is known in this direction. Recently, a non-commutative analog of the classical Banach Principle, on convergence of sequences of measurable functions generated by a sequence of linear maps on L p -spaces, was established in (3). It is expected that, as in the commuta- tive case, this principle will be instrumental in obtaining various convergence results for the averages in non-commutative setting. In (8), an individual er- godic Theorem for subsequences was proved, where the proof was based on application of the \commutative" Banach Principle. In this paper we use the Ergodic Theorem of Yeadon (11) together with the results of (3), adjusted to the bilateral almost uniform convergence, to show that the main result of (8) also holds in the vNa setting. 1. Preliminaries. Let M be a von Neumann algebra (vNa) acting on a Hilbert space H. Let I be the unit of M, and let be a faithful normal seminite trace on M. Denote by P (M) the complete lattice of all projec- tions in M. Let A(M) be the set of all closed operators aliated with M. An operator x2 A(M) is said to be -measurable if for each " > 0 there
