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13 votes
1 answer
3k views

Does this metric have an official name? Lévy metric? Ky Fan metric?

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is $$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$ if $X$ and $Y$ take values in the a ...
Jason Rute's user avatar
  • 6,287
5 votes
2 answers
642 views

Is the Hausdorff metric on sub-$\sigma$-fields separable?

Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ ...
Jason Rute's user avatar
  • 6,287
4 votes
2 answers
374 views

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

I'm reading a proof of below theorem from this paper. Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
Analyst's user avatar
  • 657
2 votes
1 answer
89 views

Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances

Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$: $$ d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|. $$ Let $\rho$ be the Levy-Prokhorov metric on the ...
ssss nnnn's user avatar
  • 177
1 vote
1 answer
141 views

Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space?

For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we ...
Rex Lee's user avatar
  • 13
0 votes
0 answers
45 views

Skorohod Space with $J_1$ topology homeomorphic to Frechet Space

Is the Skorohod space $D([0,T];\mathbb{R}^d)$ equipped with the $J_1$ topology homoeomorphic to a separable Fr\'{e}chet space. In particular, is it homeomorphic to $L_{\mu}^1(\mathcal{B}([0,1])$ ...
ABIM's user avatar
  • 5,405