## Representation Theorem

14,000,000 Leading Edge Experts on the ideXlab platform

## Contact Leading Edge Experts & Companies

The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform

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

• ##### a Representation Theorem for archimedean quadratic modules on rings
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
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.