Doubts about $D[0,1]$ set

measure-theoryprobability theoryskorohod-spaceweak-convergence

Let $D[0,1]$ be the space of real functions $f$ on $[0,1]$ that are right continuous and have left limit. That is space of cadlag functions.

Consider $D_n = \{x \in D[0,1] | x \ \mbox{has at most $n$ jumps}\}.$

My questions are

  1. Is $D_n$ closed, nowhere dense $\forall n$?
  2. If we take exactly $n$ jumps instead of at most $n$ jumps. Then can we say that $D_n$ is closed?

Note: underline topology is Skorohod topology.

I'm thinking about rational approximation of the functions from $D$. If that is possible somehow.

Best Answer

Preliminary definition. Define $\|f-g\|_I=\sup_{t\in I} \left|f(t)-g(t)\right|$ and $$d_I(f,g)=\inf_{\lambda\in \Lambda} \left\{ \|\lambda-I\|_I\vee \|f\circ\lambda - g\|_I \right\}.$$

$2.$ No. Consider the sequence $g_N\in D_1$ defined as $g_N(x)=f(x)+\frac{1}{N} 1_{\left\{x\geq t\right\}}$, with $f\in\mathcal{C}\left[0,1\right]=D_0$. It converges to $f\in D_0$.

$1.$ Yes. You can show that the complement $D^{c}_n$ (set with at least $n+1$ jumps) is open for all $n$ via induction on the number of jumps $n$:

i) [$n=0$] Choose any $g\in D^{c}_{0}$. Then, $B_{J/3}(g)\cap \mathcal{C}\left[0,1\right]=\emptyset$, where $J>0$ is the jump size of $g$. In other words, $D^{c}_{0}$ is open (or $D_0$ is closed);

ii) [$n-1\Rightarrow n$] Assume that $D_{n-1}^{c}$ is open (induction hypothesis). Further, assume that $D_n^{c}$ is not open. Then, there exists $g\in D_{n}^{c}$ and a sequence $g_N\in D_n$ with $d(g_N,g)\longrightarrow 0$. Define $I\overset{\Delta}=\left(t'-\delta,t'+\delta\right)$, where $t'$ is the instant of one of the jumps of $g$ and $\delta<\min\limits_{i=1,\ldots,M} t_{i+1}-t_i$, with $t_1,t_2,\ldots,t_M$ being the jump instants of $g$. We have that, $d(g_N,g)\longrightarrow 0 \Longrightarrow d_{I^c}(g_N,g)\longrightarrow 0$. Now, consider the map $\widetilde{g}$ defined as: $\widetilde{g}(x)=g(x)$ for all $x\in \left[0,1\right]\setminus I$ and $\widetilde{g}$ is the linear interpolation between $g(t-\delta)$ and $g(t+\delta)$ over $I$. Note that $\widetilde{g}\in D_{n-1}$ (basically, one of its jumps has been replaced by a linear interpolation). Consider that each $\widetilde{g}_N$ is obtained from $g_N$ following the same construction. Remark further that $\widetilde{g}_N\in D_{n-1}$ eventually: otherwise, we would not have $d(g_N,g)\longrightarrow 0$. We can see that $d(\widetilde{g}_N,\widetilde{g})\longrightarrow 0$ which contradicts the induction hypothesis.

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