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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. …

There is a bound, the coupling $(\xi, a \xi)$ leads to the upper bound \begin{align*} W_p(\xi, a \xi)^p &\leq \mathbb{E}[|\xi - a\xi|^p] = |(1-a)|^p \mathbb{E}[|\xi|^p]\\ \Rightarrow ~~~~ W_p(\xi, a\x …

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 …

I think I found an elementary proof of Question 2/3 for arbitrary probability distributions. In fact, it is not required that the components in the sums are squares, but general i.i.d. non-negative ra …

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 …

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 …

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 …

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 …