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Tatsuji Kawai - One of the best experts on this subject based on the ideXlab platform.
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a continuity principle equivalent to the monotone pi 0 _ 1 π 1 0 fan theorem
Archive for Mathematical Logic, 2019Co-Authors: Tatsuji KawaiAbstract:The strong continuity principle reads “every pointwise continuous function from a complete Separable Metric Space to a Metric Space is uniformly continuous near each compact image.” We show that this principle is equivalent to the fan theorem for monotone $$\varPi ^{0}_{1}$$ bars. We work in the context of constructive reverse mathematics.
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a continuity principle equivalent to the monotone pi 0 _ 1 fan theorem
arXiv: Logic, 2018Co-Authors: Tatsuji KawaiAbstract:The strong continuity principle reads "every pointwise continuous function from a complete Separable Metric Space to a Metric Space is uniformly continuous near each compact image." We show that this principle is equivalent to the fan theorem for monotone $\Pi^{0}_{1}$ bars. We work in the context of constructive reverse mathematics.
Randal Douc - One of the best experts on this subject based on the ideXlab platform.
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consistency of the maximum likelihood estimator for general hidden markov models
Annals of Statistics, 2011Co-Authors: Randal Douc, Eric Moulines, Jimmy Olsson, Ramon Van HandelAbstract:Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete Separable Metric Space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state Space models, as well as general results on linear Gaussian state Space models and finite state models. A novel aspect of our approach is an information-theoretic technique for proving identifiability, which does not require an explicit representation for the relative entropy rate. Our method of proof could therefore form a foundation for the investigation of MLE consistency in more general dependent and non-Markovian time series. Also of independent interest is a general concentration inequality for V-uniformly ergodic Markov chains.
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Consistency of the maximum likelihood estimator for general hidden Markov models
Annals of Statistics, 2011Co-Authors: Randal Douc, Eric Moulines, Jimmy Olsson, Ramon Van HandelAbstract:Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete Separable Metric Space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state Space models, as well as general results on linear Gaussian state Space models and finite state models. A novel aspect of our approach is an information-theoretic technique for proving identifiability, which does not require an explicit representation for the relative entropy rate. Our method of proof could therefore form a foundation for the investigation of MLE consistency in more general dependent and non-Markovian time series. Also of independent interest is a general concentration inequality for V-uniformly ergodic Markov chains.
Maxwell B Stinchcombe - One of the best experts on this subject based on the ideXlab platform.
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skorohod s representation theorem for sets of probabilities
Research Papers in Economics, 2014Co-Authors: Martin Dumav, Maxwell B StinchcombeAbstract: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
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skorohod s representation theorem for sets of probabilities
Social Science Research Network, 2013Co-Authors: Martin Dumav, Maxwell B StinchcombeAbstract: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.
Kawai Tatsuji - One of the best experts on this subject based on the ideXlab platform.
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A continuity principle equivalent to the monotone $\Pi^{0}_{1}$ fan theorem
2018Co-Authors: Kawai TatsujiAbstract:The strong continuity principle reads "every pointwise continuous function from a complete Separable Metric Space to a Metric Space is uniformly continuous near each compact image." We show that this principle is equivalent to the fan theorem for monotone $\Pi^{0}_{1}$ bars. We work in the context of constructive reverse mathematics.Comment: 13 page
Ramon Van Handel - One of the best experts on this subject based on the ideXlab platform.
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consistency of the maximum likelihood estimator for general hidden markov models
Annals of Statistics, 2011Co-Authors: Randal Douc, Eric Moulines, Jimmy Olsson, Ramon Van HandelAbstract:Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete Separable Metric Space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state Space models, as well as general results on linear Gaussian state Space models and finite state models. A novel aspect of our approach is an information-theoretic technique for proving identifiability, which does not require an explicit representation for the relative entropy rate. Our method of proof could therefore form a foundation for the investigation of MLE consistency in more general dependent and non-Markovian time series. Also of independent interest is a general concentration inequality for V-uniformly ergodic Markov chains.
