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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\...

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 ...

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 ...