# Questions tagged [skorohod-space]

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

47 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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