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Questions tagged [skorohod-space]

For question about Skorohod space, the space of functions which are right continuous and have left limits at each point.

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Does convergence in distribution imply convergence of regular versions of the conditional expectation?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,d)$ be a locally compact complete separable metric space $D([0,\infty),E):=\left\{x:[0,\infty)\to E\mid x\text{ is càdlàg}\right\}$...
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Given a càdlàg process $X$ and a measure μ on the Skorohod space, how can we show that μ is the law of $X$ up to the dependence on the initial value?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,d)$ be a locally compact complete separable metric space $D([0,\infty),E):=\left\{x:[0,\infty)\to E\mid x\text{ is càdlàg}\right\}$...
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Show that a càdlàg function is uniformly right-continuous on compact intervals

Let $(E,d)$ be a locally compact separable metric space, $I\subseteq\mathbb R$ be an interval, $f:I\to E$ be càdlàg and $a,b\in I$ with $a<b$. How can we show that $\left.f\right|_{[a,\:b]}$ is ...
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If $f\in C_0$ and $\lambda>0$, how can we show that $x\mapsto\int_0^\infty e^{-\lambda t}f(x(t))\:{\rm d}t$ is continuous wrt the Skorohod topology?

Let $(E,d)$ be a locally compact separable metric space, $C_0(E)$ denote the space of continuous function from $E$ to $\mathbb R$ vanishing at infinity equipped with the supremum norm and $D([0,\infty)...
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Poisson process on Skorokhod's space

For each $n=1,2,\ldots $, let $\ \xi_{n1},\ldots, \xi_{nn}$ be random and independent variables such that $\mathbb{P}(\xi_{ni}=1)=p_n \ \ $ and $\ \ \mathbb{P}(\xi_{ni}=0)=1-p_n$. Let consider the ...
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36 views

Equivalent conditions for relative compactness in the Skorohod space

Let $(E,d)$ be a complete separable metric space and $\mathcal X$ be a family of càdlàg functions $E\to[0,\infty)$. Consider the following claim: For all $t\in[0,\infty)\cap\mathbb Q$, there is a ...
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I'm trying to understand a weak convergence result for Feller processes in Ethier and Kurtz

Let $E$ be a locally compact separable metric space, $D([0,\infty),E)$ denote the Skorohod space, $C_0(E)$ denote the space of continuous functions $E\to\mathbb R$ vanishing at infinity and $E^\ast:=E\...
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If $X^n$ is a càdlàg process in a locally compact space $E$, show that $\{X^n\}$ is relatively compact as processes in the compactification of $E$

Let $E$ be a locally compact separable$^1$ metric space, $D([0,\infty),E)$ denote the Skorohod space, $C_0(E)$ denote the space of continuous functions $E\to\mathbb R$ vanishing at infinity and $E^\...
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If $E$ is a locally compact metric space and $f:[0,∞)→E$ is càdlàg, is $f$ càdlàg as a function into the one-point compactification of $E$ too?

Let $(E,\tau)$ be a locally compact Hausdorff space, $\infty\not\in E$ be an abstract point, $E^\ast:=E\uplus\left\{\infty\right\}$ and $\tau^\ast:=\tau\cup\left\{E\setminus K\cup\left\{\infty\right\}:...
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If $(Y_r)_{r∈[0,\:t]}$ is a càdlàg process, is there some relation between the distribution of $(Y_r)_{r∈[0,\:s]}$ and $(Y_r)_{r∈[0,\:t]}$ for $s≤t$?

Let $t>0$, $E$ be a separable metric space and $$D([0,t],E):=\left\{x:[0,t]\to E\mid x\text{ is càdlàg}\right\}$$ be equipped with the Skorohod topology. Now, let $(\Omega,\mathcal A,\...
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Random time change for a Poisson process and convergence with respect to the Skorohod topology

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\left(Y^{(n)}_k\right)_{k\in\mathbb N_0}$ be a time-homogeneous Markov chain on $(\Omega,\mathcal A,\operatorname P)$ and $$X^{(n)}_t=...
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Representation of the Skorokhod $J_1$-topology as a projective limit

Let $E$ be a Polish space and consider the Skorokhod $J_1$-topology on the space of cadlag functions $D_T := D([0, T], E)$ - this is just the "common" topology on this space as used in the theory of ...
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Convergence of jump function with respect to the Skorokhod metric

Consider the space $D := D([0, 1], \mathbb{N})$ of cadlag functions $f : [0, 1] \to \mathbb{N}$ equipped with the Skorokhod $J_1$-metric $d(f,g) := \inf_{\lambda \in \Lambda} \max \{ \lVert \lambda - ...
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weak convergence, relative compactness

I am interested in weak convergence of stochastic processes with sample paths in $D_{\mathbb{H}}[0,1]$. Let $\mathbb{H}$ denote a separable Hilbert space and $D_{\mathbb{H}}[0,1]$ the space of ...
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Dose Skorokhod metric generate a norm on Skorokhod space

$0$ is a constant RCLL function. So for any RCLL function $f$, we can define $|f| = d(f,0)$, where $d$ is the Skorokhod metric. Is this a norm? If not, why? If yes, why can't I search anything ...
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Borel-$\sigma$-algebra on D[0,1]

Let $\mathcal{D}:=\{\pi_{t_1,\ldots, t_d}^{-1}(B) : {t_1,\ldots, t_d} \in [0,1]$ and $B \in \mathcal{B}(\mathbb{R}^d)\}$ and $\mathcal{D}_0:=\{\pi_{t}^{-1}(B) : t\in [0,1]$ and $B \in \mathcal{B}(\...
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Is the space of convex functions a Polish space?

I'm working with continuous, increasing, convex functions (call the set $\mathcal{U}$) and viewing them as a subspace of $\mathcal{C}=\mathcal{C}([0,1])$ (perhaps this is not the best topology, ...
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Coordinate projections in D[0,1] continuous

In an older article (see Are coordinate projections in the Skorokhod space continuous?) the proof is given that the projections are continuous for x=0 and x=1. How can I conclude from $d(f,g)\geq\max\{...
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Problem with an integral equation taken from a paper

I am reading a paper (the 2015 paper by A. Falkowsy and L. Slominski Stochastic Differential Equation with Constraints Driven by Processes with Bounded $p-$variation, page 353, proof of the Lemma 3.1) ...
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Inequality involving $p-$variation and $\sup$ norms.

Let's call $\mathcal D(\Bbb R_{\ge0},\Bbb R^d)$ the Skohrokod space, i.e. the space of the functions $f:\Bbb R_{\ge0}\to\Bbb R^d$ continous on the right which admit limit on the left. Let's fix $y,l\...
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Will weak convergence of continuous functions give a continuous function?

Assume that $P_n$ is a sequence of probability measures on $D[0,1]$ with the Skorohod topology. Assume that they converge weakly to a measure $P$. Let $C$ be the space of continuous functions on $[0,1]...
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Skorohod topology extended to whole real line

Results about the standard Skorohod topology on the space $D([0,\infty))$ of cadlag functions from $[0,\infty)$ to $\mathbb{R}^d$, can easily be found in many classic texts, such as Billingsley's ...
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Skorokhod's representation Theorem: What is the filtration on the common probability space?

Assume that we have a sequence of stochastic processes $\{X_n\}$ and a process $X$ whose trajectories belong to the space $D([0, T],\mathbb{R})$ of right-continuous, having left limit functions $ \...
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122 views

Weak convergence of stochastic process and random times

Suppose that we have real-valued cadlag stochastic processes $X_n=(X_n(t))_{t\in\mathbb{R}}$ (for $n\in\mathbb{N}$), and $X=(X(t))_{t\in\mathbb{R}}$, such that as $n\rightarrow\infty$ $\quad X_n$ ...
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Sequences of SDEs

I've been looking for a while an cannot seem to find the answer to this question: If $Y_n$ is a sequenence of semi-martingales solving the sequence of stochastic differential equations: $$ dY_n(t) = \...
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Criterion for relative compactness in uniform spaces

I am having problems in understanding a criterion for relative compactness given in a book (see below for details if you are interested) on SPDEs. However, I think it just invokes a pretty general ...
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A necessity property of compact subset of Skorokhod space

This is the Problem 3.16 of Ethier and Kurtz's Markov Processes Characterization and Convergence Let $(E,r)$ be complete. Show that if $A$ is compact in $D_E[0,\infty)$ (the skorokhod space with ...
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262 views

Tightness of (sum of) elements of the Skorokhod space

Assume $(x_n)_{n\geq1}, (y_n)_{n\geq1}$ are sequences of elements of the Skorokhod space $\mathcal D = \mathcal D(\mathbb R)$ of cadlag functions with range in $\mathbb R$, endowed with the usual $...
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Space of cadlag functions - Nonexistence of a TVS Polish topology?

Consider the space $D := D([0,1], \mathbb{R})$ of real-valued cadlag functions on $[0,1]$. On $D$ one can invent several (more or less) well-known topologies: the uniform topology $U$: $(D, U)$ is a ...
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361 views

Skorokhod space with uniform norm is Banach

Let $D := D([0,t])$ be the Skorokhod space of right-continuous functions with left limits taking values in $\mathbb{R}^d$. Equip $D$ with the supremum norm $||f||_\infty = \sup_{s \in [0,t]}|f(s)|$. ...
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89 views

Compact set on functions space

Let $(D[0,T], X\times X)$ the set of cadlag functions from $[0,T]$ to $X\times X$. If I have a compact subset $K$ in $(D[0,T], X)$ and another compact subset $H$ in $(D[0,T], X)$, is $K\times H$ a ...
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394 views

Convergence in “the” Skorohod topology for monotone functions

Let $x_n$, $x$ be nondecreasing cadlag functions on $[0,T]$. To get $x_n\to x$ in Skorokhods $M_1$-topology, we only have to prove convergence on a dense subset of $[0,T]$ including 0. (see Whitt: ...
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Continuous function on the Skorohod space

I have a process $(X,Y)\in D([0,T],\mathbb{R}^2)$ where $D([0,T],\mathbb{R}^2)$ is the set of cadlag functions with Skorohod metric. Let $A=\{\sup_{t\in[0,T]}|X(t)-\int_0^tX(s)Y(s)ds|>\epsilon\}$ ...
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Sub-additivity of the càdlàg continuity modulus

Let $f \colon [0,1] \to \mathbb{R}$ be a function and define $$\varpi'_f(\delta) = \inf_{\{t_i\}} \max_{i=1,\dots,n} \sup_{t,s \in [t_{i-1},t_{i})}|f(t)-f(s)|$$ where the infimum is taken over all ...
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non-separable metric space and measurablility of its elements

I'm studying Skorokhod space, which consists of cadlag functions, and I encountered the following sentence: If a metric space $(\mathbb{S}, \mathcal{S}, d)$ is not separable, then functions that ...
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complete proof of non-separability of D space

Let $D$ be a set of càdlàg functions on $[0,1]$. Define $f_\alpha(\cdot) \equiv 1(\cdot \ge \alpha)$ for $\alpha \in [0,1]$, which is obviously in $D$. Then, if we denote $|| \cdot ||$ as a uniform ...
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A basic question on convergence in Skorohod metric

Consider a sequence of piecewise constant (positive) function in the space $D$ with Skorohod topology such that each of the functions is zero at the origin and any two discontinuities are at least $\...
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A question about the Skorokhod topology

I have a question which may be naive but I can not find the answer in general reference about Skorokhod topology. Let $\{w_n\}_{n\ge 0}$ be a sequence of cadlag functions defined on $[0,1]$ such ...
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1answer
600 views

Weak convergence with respect to the uniform topology on cadlag functions

Suppose I have a random sequence $X_n$ of cadlag functions on $[0,1]$ that converge weakly to $X$. In general this is meant with respect to the Skorkhod metric but suppose here I have that $X$ is ...
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322 views

Skorohod convergence (space of right continuous functions with left limit)

If $f_n$ is a sequence of functions of the Skorohod Space $D([0,\infty),E)$, where $E$ is a separable Banach space, such that $f_n \to f$ in the Skorohod topology. Is it possible that there exists a $...
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801 views

Borel $\sigma$-algebras on the Skorohod space $D[0,1]$

On the Skorohod space $D[0,1]$ of cadlag functions one usually considers either the uniform norm $\|\cdot\|_{\infty}$ or the $J_1$-metric $\varrho$. I was wondering whether both generate the same ...
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397 views

Skorokhod convergence VS uniform convergence

Skorohod convergence does not imply uniform convergence. Billingsley quotes a counterexample: for $0\leq\alpha<1$ the sequence $x_n(t)=1_{[0,\alpha +\frac{1}{n})}(t)$ does not converge uniformly to ...
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183 views

Question about the relation between the Skorohod norm and the uniform norm

Is it true that convergence of random elements in $D[0,1]$, equipped with the Skorohod norm, to a limit with continuous paths (i.e., a limit in $C[0,1]$) implies that the same convergence holds with ...
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221 views

Relative sizes of Skorokhod and product topologies on space of sample paths

Let $E$ denote a compact metric space. Let $T$ denote the non-negative reals. Let $E^T$ denote the class of all functions from $T$ to $E$. Let $C$ denote the subset of $E^T$ consisting of càdlàg ...
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162 views

Skorohod space measurable function

Consider the space of RCLL (Cadlag) functions on the domain $[0,1]$ and endowed with the Skorohod topology. Let us consider the set $S := \{x: \omega_x (\delta) \leq \epsilon\}$, where $\omega_x (\...
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100 views

Computationally efficient means of determining distance in the Skorohod Topology?

I have two functions f and g in a computer. Domain 1...N. I'd like to compute their distance using the Skorohod Topology in an efficient manner. (I first ran across this metric many years ago in ...
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1answer
1k views

A proof that Skorohod metric is a metric

I'm reading Billingsley's "Convergence of probability measures" (1968), p. 111. The definitions are: $D$ - the space of $\textit{cadlag}$ functions on [0,1], $\Lambda$ - the class of strictly ...