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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 …

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 …

Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ endo …

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 …

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 …

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 …

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

I think I have an answer for the case p = 1, K = 2. I write "I think" because my computation does not coincide with the example values for $N=4$ posted earlier by OP in a comment, but I really cannot …

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 …

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 …

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 …

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 …

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 …

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 …