Representation Theorem

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Jaka Cimpric - One of the best experts on this subject based on the ideXlab platform.

  • a Representation Theorem for archimedean quadratic modules on rings
    Canadian Mathematical Bulletin, 2009
    Co-Authors: Jaka Cimpric
    Abstract:

    We present a new approach to noncommutative real algebraic geometry based on the Representation theory of C∗-algebras. An important result in commutative real algebraic geometry is Jacobi’s Representation Theorem for archimedean quadratic modules on commutative rings. We show that this Theorem is a consequence of the Gelfand–Naimark Representation Theorem for commutative C∗-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi’s Theorem to associative rings with involution. Received by the editors January 16, 2004; revised March 7, 2005. AMS subject classification: Primary 16W80; secondary 46L05, 46L89, 14P99.

  • a Representation Theorem for archimedean quadratic modules on rings
    arXiv: Rings and Algebras, 2008
    Co-Authors: Jaka Cimpric
    Abstract:

    We present a new approach to noncommutative real algebraic geometry based on the Representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's Representation Theorem for archimedean quadratic modules on commutative rings, \cite[Theorem 5]{jacobi}. We show that this Theorem is a consequence of the Gelfand-Naimark Representation Theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand-Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's Theorem to associative rings with involution.

Maxwell B Stinchcombe - One of the best experts on this subject based on the ideXlab platform.

  • skorohod s Representation Theorem for sets of probabilities
    Research Papers in Economics, 2014
    Co-Authors: Martin Dumav, Maxwell B Stinchcombe
    Abstract:

    From Breiman et al. [3], a set of probabilities, Π, on a measure space, (Ω,F), is strongly zero-one if there exists an E ∈ F , a measurable, onto φ : Ω → Π such that for all p ∈ Π, p(φ−1(p)) = 1. Suppose that Π is an uncountable, measurable, strongly zero-one set of non-atomic probabilities on a standard measure space, that M is a complete, separable metric space, ∆M is the set of Borel probabilities on M and Comp(∆M) is the class of non-empty, compact subsets of ∆M with the Hausdorff metric. There exists a jointly measurable H : Comp(∆M) × Ω → M such that for all K ∈ Comp(∆M), H(K,Π) = K, and if dρH(Kn,K0)→ 0, then for all p ∈ Π, p({ω : H(Kn, ω)→ H(K0, ω)}) = 1. When each Kn and Π are singleton sets, this is the Blackwell and Dubins [2] version of Skorohod’s Representation Theorem. 1. Extending Skorohod’s Representation Theorem to Sets Let (M,d) be a complete separable metric (Polish) space, M the Borel σ-field on M , ∆M the set of countably additive probabilities on M, and Cb(M) the continuous, R-valued functions on M . In ∆M, let ρ(·, ·) be any metric inducing the weak∗ topology, that is, ρ(μn, μ0)→ 0 iff ∫ f dμn → ∫ f dμ0 for every f ∈ Cb(M). The Borel σ-field, DM on ∆M is the smallest σ-field containing all of the ρ-open sets, and it can alternatively be characterized as the smallest σ-field containing all sets of the form {μ : μ(E) ≤ r}, E ∈M, r ∈ [0, 1]. A measurable isomorphism between the measure spaces (Ω,F) and (Ω′,F ′) is a bijection that is measurable and has a measurable inverse. A measure space (Ω,F) is called standard if it is measurably isomorphic to a Borel measurable subset of a Polish space. Let ∆F denote the set of countably additive probabilities on F , and DF the sigma-field generated by sets of the form {p ∈ ∆(F) : p(E) ≤ r}, E ∈ F , r ∈ [0, 1]. It is known that the measure space (∆F ,DF ) is standard iff (Ω,F) is ([5, Theorem III.60]. In particular, (∆M,DM) is standard. Let (Ω,F) be a standard measure space and p a non-atomic, countably additive probability on F . Skorohod [10] showed that if (M,d) is a complete separable metric space and ρ(μn, μ0)→ 0, then there exist random variables, Xn, X0 : Ω→M such that Sko(a) Xn(p) = μn, X0(p) = μ0, and Sko(b) p({ω : Xn(ω)→ X(ω)}) = 1 where Xn(p) is the image law of the distribution p under the random variable Xn, that is, Xn(p)(B) = p(X −1 n (B)) for each B ∈ M. Blackwell and Dubins Date: May 31, 2013

  • skorohod s Representation Theorem for sets of probabilities
    Social Science Research Network, 2013
    Co-Authors: Martin Dumav, Maxwell B Stinchcombe
    Abstract:

    From Breiman et al. (1964), a set of probabilities, Pi, on a measure space, (Omega,F), is strongly zero-one if there exists an E in F, a measurable, onto phi:Omega -> Pi such that for all p in Pi, p(phi^{-1}(p))=1. Suppose that Pi is an uncountable, measurable, strongly zero-one set of non-atomic probabilities on a standard measure space, that M is a complete, separable metric space, Delta_M is the set of Borel probabilities on M and Comp(Delta_M) is the class of non-empty, compact subsets of Delta_M with the Hausdor ff metric. There exists a jointly measurable H: Comp(Delta_M) x Omega ->M such that for all K in Comp(Delta_M), H(K,Pi) = K, and if d_H^rho(K_n,K_0) -->0, then for all p in Pi, p({omega: H(K_n,omega) -->H(K_0,omega)})=1. When each K_n and Pi are singleton sets, this is the Blackwell and Dubins (1983) version of Skorohod's Representation Theorem.

Martin Dumav - One of the best experts on this subject based on the ideXlab platform.

  • skorohod s Representation Theorem for sets of probabilities
    Research Papers in Economics, 2014
    Co-Authors: Martin Dumav, Maxwell B Stinchcombe
    Abstract:

    From Breiman et al. [3], a set of probabilities, Π, on a measure space, (Ω,F), is strongly zero-one if there exists an E ∈ F , a measurable, onto φ : Ω → Π such that for all p ∈ Π, p(φ−1(p)) = 1. Suppose that Π is an uncountable, measurable, strongly zero-one set of non-atomic probabilities on a standard measure space, that M is a complete, separable metric space, ∆M is the set of Borel probabilities on M and Comp(∆M) is the class of non-empty, compact subsets of ∆M with the Hausdorff metric. There exists a jointly measurable H : Comp(∆M) × Ω → M such that for all K ∈ Comp(∆M), H(K,Π) = K, and if dρH(Kn,K0)→ 0, then for all p ∈ Π, p({ω : H(Kn, ω)→ H(K0, ω)}) = 1. When each Kn and Π are singleton sets, this is the Blackwell and Dubins [2] version of Skorohod’s Representation Theorem. 1. Extending Skorohod’s Representation Theorem to Sets Let (M,d) be a complete separable metric (Polish) space, M the Borel σ-field on M , ∆M the set of countably additive probabilities on M, and Cb(M) the continuous, R-valued functions on M . In ∆M, let ρ(·, ·) be any metric inducing the weak∗ topology, that is, ρ(μn, μ0)→ 0 iff ∫ f dμn → ∫ f dμ0 for every f ∈ Cb(M). The Borel σ-field, DM on ∆M is the smallest σ-field containing all of the ρ-open sets, and it can alternatively be characterized as the smallest σ-field containing all sets of the form {μ : μ(E) ≤ r}, E ∈M, r ∈ [0, 1]. A measurable isomorphism between the measure spaces (Ω,F) and (Ω′,F ′) is a bijection that is measurable and has a measurable inverse. A measure space (Ω,F) is called standard if it is measurably isomorphic to a Borel measurable subset of a Polish space. Let ∆F denote the set of countably additive probabilities on F , and DF the sigma-field generated by sets of the form {p ∈ ∆(F) : p(E) ≤ r}, E ∈ F , r ∈ [0, 1]. It is known that the measure space (∆F ,DF ) is standard iff (Ω,F) is ([5, Theorem III.60]. In particular, (∆M,DM) is standard. Let (Ω,F) be a standard measure space and p a non-atomic, countably additive probability on F . Skorohod [10] showed that if (M,d) is a complete separable metric space and ρ(μn, μ0)→ 0, then there exist random variables, Xn, X0 : Ω→M such that Sko(a) Xn(p) = μn, X0(p) = μ0, and Sko(b) p({ω : Xn(ω)→ X(ω)}) = 1 where Xn(p) is the image law of the distribution p under the random variable Xn, that is, Xn(p)(B) = p(X −1 n (B)) for each B ∈ M. Blackwell and Dubins Date: May 31, 2013

  • skorohod s Representation Theorem for sets of probabilities
    Social Science Research Network, 2013
    Co-Authors: Martin Dumav, Maxwell B Stinchcombe
    Abstract:

    From Breiman et al. (1964), a set of probabilities, Pi, on a measure space, (Omega,F), is strongly zero-one if there exists an E in F, a measurable, onto phi:Omega -> Pi such that for all p in Pi, p(phi^{-1}(p))=1. Suppose that Pi is an uncountable, measurable, strongly zero-one set of non-atomic probabilities on a standard measure space, that M is a complete, separable metric space, Delta_M is the set of Borel probabilities on M and Comp(Delta_M) is the class of non-empty, compact subsets of Delta_M with the Hausdor ff metric. There exists a jointly measurable H: Comp(Delta_M) x Omega ->M such that for all K in Comp(Delta_M), H(K,Pi) = K, and if d_H^rho(K_n,K_0) -->0, then for all p in Pi, p({omega: H(K_n,omega) -->H(K_0,omega)})=1. When each K_n and Pi are singleton sets, this is the Blackwell and Dubins (1983) version of Skorohod's Representation Theorem.

Pietro Rigo - One of the best experts on this subject based on the ideXlab platform.

  • a skorohod Representation Theorem for uniform distance
    Probability Theory and Related Fields, 2011
    Co-Authors: Patrizia Berti, Luca Pratelli, Pietro Rigo
    Abstract:

    Let μ n be a probability measure on the Borel σ-field on D[0, 1] with respect to Skorohod distance, n ≥ 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables X n such that X n ~ μ n for all n ≥ 0 and ||X n − X 0|| → 0 in probability, where ||·|| is the sup-norm. Such conditions do not require μ 0 separable under ||·||. Applications to exchangeable empirical processes and to pure jump processes are given as well.

  • a skorohod Representation Theorem for uniform distance
    Quaderni di Dipartimento, 2010
    Co-Authors: Patrizia Berti, Luca Pratelli, Pietro Rigo
    Abstract:

    Let µn be a probability measure on the Borel sigma-field on D[0, 1] with respect to Skorohod distance, n = 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables Xn such that Xn tilde µn for all n = 0 and ||Xn - X0|| --> 0 in probability, where ||·|| is the sup-norm. Such conditions do not require µ0 separable under ||·||. Applications to exchangeable empirical processes and to pure jump processes are given as well.

  • skorohod Representation Theorem via disintegrations
    Quaderni di Dipartimento, 2009
    Co-Authors: Patrizia Berti, Luca Pratelli, Pietro Rigo
    Abstract:

    Let (µn : n >= 0) be Borel probabilities on a metric space S such that µn -> µ0 weakly. Say that Skorohod Representation holds if, on some probability space, there are S-valued random variables Xn satisfying Xn - µn for all n and Xn -> X0 in probability. By Skorohod’s Theorem, Skorohod Representation holds (with Xn -> X0 almost uniformly) if µ0 is separable. Two results are proved in this paper. First, Skorohod Representation may fail if µ0 is not separable (provided, of course, non separable probabilities exist). Second, independently of µ0 separable or not, Skorohod Representation holds if W(µn, µ0) -> 0 where W is Wasserstein distance (suitably adapted). The converse is essentially true as well. Such a W is a version of Wasserstein distance which can be defined for any metric space S satisfying a mild condition. To prove the quoted results (and to define W), disintegrable probability measures are fundamental.

M.m. Carroll - One of the best experts on this subject based on the ideXlab platform.

  • A Representation Theorem for volume-preserving transformations
    International Journal of Non-Linear Mechanics, 2004
    Co-Authors: M.m. Carroll
    Abstract:

    Abstract A general solution is presented for the partial differential equation ∂u / ∂x = k ( x ), where u and x are n -vector fields, ∂u / ∂x denotes the Jacobian of the transformation x → u and k ( x ) is a scalar-valued function. The solution for the case k ( x )=1 is of special interest because it furnishes a Representation Theorem for volume-preserving transformations in an n -dimensional space. Such a Representation for the case n =2 was obtained by Gauss. The solution for n =3, presented here, furnishes a Representation for isochoric (volume-preserving) finite deformations, which are important in the mechanics of highly deformable incompressible solid materials.