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It looks to me like the authors just wanted to put in some ridiculous values which "are clearly sufficient" so that they don't have to work out the details. The choice $\eta = c_1^{1/10}$ should ensur …

Take $\pi^N$ with $AW(\pi^N, \pi) \leq \frac{1}{N}$, where we denote by $\mu^N$ and $\nu^N$ the marginals of $\pi^N$. Note that by the backward induction for $AW$ (cf. here), it holds $$ AW(\pi, \pi^N …

As already mentioned in the comments, this is really a standard result, see, e.g., Lemma 3.1. in Kallenberg's book.

I find the $d$-separation criterion (see, e.g., Theorem 2 here; note however the preceding definition, which basically means we are treating discrete random variables) a really useful sufficient crite …

Perhaps another simple argument that $\mathcal{X}$ is indeed equal to $\mathcal{W}_2(\mathbb{R}^d$): Starting from the fact that there is a bimeasurable bijection $b : \mathbb{R}^d \rightarrow \mathbb …

The following question came up when trying to numerically solve some variants of the Gromov-Wasserstein distance. Setting: Let $(X_1, d_1), (X_2, d_2)$ be two compact, separable and complete metric s …

For $n\in\mathbb{N}$, denote by $\mathcal{Q}_n$ the set of all probability measures on $\mathbb{R}$ that are supported on at most $n$ points. Question: Is it known which element in $\mathcal{Q}_n$ is …

There are various papers where this question occurs. I guess a paper which directly covers the case you are interested in is https://arxiv.org/pdf/1901.07407.pdf . Note that here, the marginals don't …

Just a partial answer, but the proposed inequality doesn't hold. Take $p = [0.2, 0.8], q = [0.1, 0.9]$. Then $H(p) = 0.2 \log(5) + 0.8 \log(1/0.8) \approx 0.5$, $H(q) = 0.1 \log(10) + 0.9 \log(1/0. …

Let $E$ be a polish space, $\mathcal{P}(E)$ the Borel probability measures on $E$ with the topology of weak convergence and $\mathcal{Q} \subset \mathcal{P}(E)$ a convex and compact set. First, the m …

That's false even for $n=3$. Denote by $B(p)$ the Bernoulli distribution which has probability $p$ of being 1 and $(1-p)$ to be 0. Define the three Bernoulli variables $(X_1, X_2, X_3)$ by $ X_1 \si …

The 2013 paper "A Class of Probability Distributions that is Closed with Respect to Addition as Well as Multiplication of Independent Random Variables" by Lennard Bondesson is closely related to your …

We are on $\mathbb{R}^d$ with Borel $\sigma$-algebra. Let $\mu_1, ..., \mu_d$ be probability measures on $\mathbb{R}$ and $\Pi(\mu_1, \mu_2, ..., \mu_d)$ be the set of probability measures on $\mathbb …

Maybe to add to the point of calculating $\max_{P_\varepsilon} \mathbb{E}_{P_\varepsilon}[f]$: I will write this a bit more in line with the literature I will refer to. Let $(X, d)$ be some polish spa …

It doesn't hold. Note that if one chooses $P= \frac{1}{n^2} \delta_n + \frac{n^2-1}{n^2} \delta_x$ with $x$ chosen such that $\mu = 1$, i.e. $x = \frac{n^2-n}{n^2-1}$, then $\mu_3 \sim n$ asymptotical …