All Questions
7 questions
25
votes
6
answers
10k
views
Metrization of weak convergence of signed measures
Edit: Changed from "Hausdorff" to "metric" spaces.
Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, ...
- 12.7k
5
votes
0
answers
266
views
Concentration inequalities for random measures
For random variables $X_1,\dots,X_n$ with common mean $\mathbb{E}[X_i]=\mu$ and common bounds $a\leq X_i\leq b$, we have the very useful Hoeffding's inequality:
$$\mathbb{P}\left(\left|\mu -\frac1n\...
- 101
4
votes
2
answers
255
views
Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
- 825
1
vote
1
answer
215
views
Compactness with respect to topology induced by total-variation distance
I've been working on a problem and at some point in the proof I need to show that the following set $$\left\{\mu \in \mathcal{P}_{ac}(\mathbb R^d): \int \varphi(x)\mu(\mathrm{d}x)\leq C\right\}$$
is ...
- 21
1
vote
0
answers
45
views
Modifiying a sequence of measures to assign a certain value when integrating a fixed function?
Let $f:\mathbb R ^d\to \mathbb R$ be some continuous function, $|f|\leq A(1+|x|)$, where $|\cdot|$ denotes the usual Euclidean norm. Fix a measure $\mu$ and constant $C$.
Assume that $\mu_n$ is a ...
- 291
0
votes
1
answer
207
views
Some continuity issues of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
- 54
0
votes
0
answers
114
views
Some stability and estimate of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
- 54