Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for distribution-valued processes (ProjectEuclid link) for its definition. Denote by $\rho$ the corresponding metric. I have the questions as follows :

  1. Is the space $\big(D(0,T),\rho\big)$ separable?

  2. Let $(\mathbb P_n)_{n\ge 1}$ be a weakly convergent sequence of probability measures on $D(0,T)$ endowed with $\rho$. Denote by $\mathbb P$ its limit. Is there a dense subset $I\subset [0,T]$ s.t.

$$\mathbb P_n\circ \phi_t^{-1} \stackrel{n\to\infty}{\longrightarrow} \mathbb P\circ \phi_t^{-1},\quad \quad \quad \mbox{for all } t\in I,$$

where $\mathbb P_n\circ \phi_t^{-1}$ denotes the image measure of $\mathbb P_n$ under $\phi_t$ and $\phi_t:D(0,T)\to\mathbb R$ is defined by $\phi_t(f):=f(t)$ for any $f\in D(0,T)$.

I am unable to find the related references. Any answers or references are highly appreciated.

PS: The Skorokhod M1 topology is not the Skorokhod J1 topology that is used mostly. This topology M1 is strictly weaker than J1, and is defined by the so called parametric representation of graphs.


The answer to both questions is positive. You can find the answers in the book Convergence of Probability Measures by P. Billingsley.

  1. Separability is proved in page 112 of the book (in Section 14).

  2. The set $I$ can be chosen to be the set of times $t$ such that the random function is continuous at $t$, $\mathbb P$-almost surely. See the beginning of Section 15 of the above book, which proves that the complement of this set is at most countable. Also, Theorem 15.1 is a converse to your problem.

  • $\begingroup$ Thanks for your reply. I do not find the results for the Skorokhod M1 topology, while Billingsley's book is mostly concerned with the Skorokhod J1 topology. Could you please be more precise? The book of Witt columbia.edu/~ww2040/preflongno.pdf is indeed about the M1 topology (Chapter 12), but there is no free version online $\endgroup$ – Neymar Sep 12 '20 at 12:49
  • $\begingroup$ You are right, I thought you are asking about the topology in Billingsley's book. $\endgroup$ – Ali Khezeli Sep 13 '20 at 5:00
  • $\begingroup$ @Neymar Since M1 is weaker than J1, any countable set that is dense for J1 is also dense for M1. $\endgroup$ – Martin Hairer Oct 12 '20 at 13:29

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