Skorohod Representation Theorem Research Articles

A Strong Version of the Skorohod Representation Theorem

For each nge 0, let mu _n be a tight probability measure on the Borel sigma -field of a metric space S. Let (T,{mathcal {C}}) be a measurable space such that the diagonal bigl {(t,t):tin Tbigr } belongs to {mathcal {C}}otimes {mathcal {C}}. Fix a measurable function g:Srightarrow T and suppose mu _n=mu _0 on g^{-1}({mathcal {C}}) for all nge 0. Necessary and sufficient conditions for the existence of S-valued random variables X_n, defined on the same probability space and satisfying Xn⟶a.s.X0,Xn∼μnandg(Xn)=g(X0)for alln≥0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} X_n\\overset{\ ext {a.s.}}{\\longrightarrow }X_0,\\quad X_n\\sim \\mu _n\\,\ ext { and } \\,g(X_n)=g(X_0)\\,\ ext { for all }n\\ge 0, \\end{aligned}$$\\end{document}are given. Such conditions are then applied to several examples. The tightness condition on mu _0 can be dropped at the price of some assumptions on S and g.

Domains and stochastic processes

Domain theory has a long history of applications in theoretical computer science and mathematics. In this article, we explore the relation of domain theory to probability theory and stochastic processes. The goal is to establish a theory in which Polish spaces are replaced by domains, and measurable maps are replaced by Scott-continuous functions. We illustrate the approach by recasting one of the fundamental results of stochastic process theory – Skorohod's Representation Theorem – in domain-theoretic terms. We anticipate the domain-theoretic version of results like Skorohod's Theorem will improve our understanding of probabilistic choice in computational models, and help devise models of probabilistic programming, with its focus on programming languages that support sampling from distributions where the results are applied to Bayesian reasoning.

Local martingale solutions to the stochastic two layer shallow water equations with multiplicative white noise

We study the two layers shallow water equations on a bounded domain M⊂R2 driven by a multiplicative white noise, and obtain the existence and uniqueness of a maximal pathwise solution for a limited period of time, the time of existence being strictly positive almost surely. The proof makes use of anisotropic estimates and stopping time arguments, of the Skorohod representation theorem, and the Gyöngy–Krylov theorem which is an infinite dimensional analogue of the Yamada–Watanabe theorem.

Skorohod's Representation Theorem and Optimal Strategies for Markets with Frictions

We prove the existence of optimal strategies for agents with cumulative prospect theory preferences who trade in a continuous-time illiquid market, transcending known results which previously pertained only to risk-averse utility maximizers. The arguments exploit an extension of Skorohod's representation theorem for tight sequences of probability measures. This method is applicable in a number of similar optimization problems.

Local martingale solutions to the stochastic one layer shallow water equations

We consider the single layer shallow water equations on a bounded domain M⊂R2 forced by a multiplicative white noise, and obtain the existence and uniqueness of a maximal pathwise solution for a short period of time. The proof relies on the Skorohod representation theorem, the Gyöngy–Krylov theorem, stopping time arguments, and isotropic estimates.

Convergence in density in finite time windows and the Skorohod representation

According to the Dudley-Wichura extension of the Skorohod representation theorem, convergence in distribution to a limit in a separable set is equivalent to the existence of a coupling with elements converging a.s.in the metric. A density analogue of this theorem says that a sequence of probability densities on a general measurable space has a probability density as a pointwise lower limit if and only if there exists a coupling with elements converging a.s.in the discrete metric. In this paper the discrete-metric theorem is extended to stochastic processes considered in a widening time window. The extension is then used to prove the separability version of the Skorohod representation theorem. The paper concludes with an application to Markov chains.

Gluing lemmas and Skorohod representations

Let $(\mathcal{X},\mathcal{E})$, $(\mathcal{Y},\mathcal{F})$ and $(\mathcal{Z},\mathcal{G})$ be measurable spaces. Suppose we are given two probability measures $\gamma$ and $\tau$, with $\gamma$ defined on $(\mathcal{X}\times\mathcal{Y},\mathcal{E}\otimes\mathcal{F}$ and $\tau$ on $(\mathcal{X}\times\mathcal{Z},\mathcal{E}\otimes\mathcal{G})$. Conditions for the existence of random variables $X,Y,Z$, defined on the same probability space $(\Omega,\mathcal{A},P)$ and satisfying $$(X,Y)\sim\gamma\,\text{ and }\,(X,Z)\sim\tau,$$ are given. The probability $P$ may be finitely additive or $\sigma$-additive. As an application, a version of Skorohod representation theorem is proved. Such a version does not require separability of the limit probability law, and answers (in a finitely additive setting) a question raised in preceding works.

Comparative and qualitative robustness for law-invariant risk measures

When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel’s classical notion of qualitative robustness is not suitable for risk measurement, and we propose and analyze a refined notion of robustness that applies to tail-dependent law-invariant convex risk measures on Orlicz spaces. This concept captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for ψ-weak convergence.

Skorohod's Representation Theorem for Sets of Probabilities

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.

A Skorohod representation theorem without separability

Let $(S,d)$ be a metric space, $\mathcal{G}$ a $\sigma$-field on $S$ and $(\mu_n:n\geq 0)$ a sequence of probabilities on $\mathcal{G}$. Suppose $\mathcal{G}$ countably generated, the map $(x,y)\mapsto d(x,y)$ measurable with respect to $\mathcal{G}\otimes\mathcal{G}$, and $\mu_n$ perfect for $n>0$. Say that $(\mu_n)$ has a Skorohod representation if, on some probability space, there are random variables $X_n$ such that<br />\begin{equation*}<br />X_n\sim\mu_n\text{ for all }n\geq 0\quad\text{and}\quad d(X_n,X_0)\overset{P}\longrightarrow 0.<br />\end{equation*}<br />It is shown that $(\mu_n)$ has a Skorohod representation if and only if<br />\begin{equation*}<br />\lim_n\,\sup_f\,\left|\mu_n(f)-\mu_0(f)\right|=0,<br />\end{equation*}<br />where $\sup$ is over those $f:S\rightarrow [-1,1]$ which are $\mathcal{G}$-universally measurable and satisfy $\left|f(x)-f(y)\right|\leq 1\wedge d(x,y)$. An useful consequence is that Skorohod representations are preserved under mixtures. The result applies even if $\mu_0$ fails to be $d$-separable. Some possible applications are given as well.

A Skorohod representation theorem for uniform distance

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.

Skorohod representation theorem via disintegrations

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.

Skorohod representation on a given probability space

Let \((\Omega,\mathcal{A},P)\) be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and \(X_n:\Omega\rightarrow S\) an arbitrary map, n = 1,2,.... If μ is tight and X n converges in distribution to μ (in Hoffmann–Jørgensen’s sense), then X∼μ for some S-valued random variable X on \((\Omega,\mathcal{A},P)\). If, in addition, the X n are measurable and tight, there are S-valued random variables \(\overset{\sim}{X}_n\) and X, defined on \((\Omega,\mathcal{A},P)\), such that \(\overset{\sim}{X}_n\sim X_n\), X∼μ, and \(\overset{\sim}{X}_{n_k}\rightarrow X\) a.s. for some subsequence (n k ). Further, \(\overset{\sim}{X}_n\rightarrow X\) a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to \(\sigma(\mathcal{A}\cup\{H\})\) for some H⊂Ω with P *(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken \(((0,1),\sigma(\mathcal{U}\cup\{H\}),m_H)\), for some H⊂ (0,1) with outer Lebesgue measure 1, where \(\mathcal{U}\) is the Borel σ-field on (0,1) and m H the only extension of Lebesgue measure such that m H (H) = 1. In order to prove the previous results, it is also shown that, if X n converges in distribution to a separable limit, then X n k converges stably for some subsequence (n k ).

On the behavior of the conditional expectations in skorohod representation theorem

In this paper we deal with the Skorohod representation of a given system of probability measures. More precisely, we give conditions for the existence of a Skorohod representation ( X,( X n )) with the following additional property: for each real number p⩾1 and each real random variable Z in L p , the conditional expectation E[ Z| X n ] converges in L p to the conditional expectation E[ Z| X].

On the behavior of the conditional expectations in the Skorohod representation theorem

On the behavior of the conditional expectations in the Skorohod representation theorem

Weak convergence of stochastic integrals driven by martingale measure

Let S′( R d) be the dual of Schwartz space, S( R d) , { M n } be a sequence of martingale measures and let F be some suitable function space such as C 0( R d) , L p( R d) , p ⩾ 2 or C m,y 0( R d) . We find conditions under which ( X n , M n ) ⇒ ( X, M) in the Skorohod topology in D F × S′( R d) [0, ∞) implies ∫ X n ( x, s) M n (d x, d s) ⇒ ∫ X( x, s) M(d x, d s) in the Skorohod topology in D S′( R d) [0, ∞) . We use the idea of regularization to reduce S′( R d) to a metrizable subspace in order to apply the Skorohod representation theorem and then appropriate the randomized mapping constructed by Kurtz and Protter to get step functions approximating the integrands. Using this result, we prove weak convergence of certain double stochastic integrals studied by Walsh. Let ∅ ∈ S( R 2d) , { η n } be a sequence of Brownian density processes and { W n } and { Z n } be two sequences of martingale measures generated by particle systems. We consider the weak convergence of ∫ ∅( x, y) η n s (d x) W n (d x, d y) and ∫ ∅( x, y) η n s (d x) Z n (d x, d y).

A review on strong convergence of weighted sums of random elements based on Wasserstein metrics

In a previous paper the authors proposed a simple method to extend results about almost sure convergence for weighted sums of real random variables to the case of Banach-valued random elements. The method arises from the extension of Skorohod's Representation Theorem for weakly convergent sequences due to Blackwell and Dubins, applied to the general framework of weakly equivalent tight sequences of probability measures. This provides a scheme which permits us to handle separately a problem that behaves like the Glivenko-Cantelli Theorem and a question on uniform integrability which generally is reduced to the real valued version of the general problem to be solved.In this paper we prove that Wasserstein's metrics can play the same role as Skorohod's Representation Theorem in the preceding scheme. We also show that our method can be applied to obtain results with respect to various summability methods (Abel, Euler, …) even in the case in which the ‘weights’ are linear operators.

Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions

The equivalence of sequences of probability measures jointly with the extension of Skorohod's representation theorem due to Blackwell and Dubins is used to obtain strong convergence of weighted sums of random elements in a separable Banach space. Our results include most of the known work on this topic without geometric restrictions on the space. The simple technique developed gives a unified method to extend results on this topic for real random variables to Banach-valued random elements. This technique is also applied to the proof of strong convergence of some statistical functionals.

A Strong Law of Large Numbers for Martingales

We derive a moment inequality for the Skorohod representation theorem and apply it to obtain a strong law of large numbers for martingales.

A strong law of large numbers for martingales

We derive a moment inequality for the Skorohod representation theorem and apply it to obtain a strong law of large numbers for martingales.