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Let's have $X,Y \in \mathbb{R}$ with probability measures $\mu, \nu$, then the Kolmogorov-Smirnov distance is defined as follows $$ d_K(X,Y)=\underset{x \in \mathbb{R}}{sup}\{|F_X(x) - F_Y(x)|\} $$

where $F_X(x)$ is the comulative distribution function of X.

If $X,Y$ have finite p-momentum then the p-Wasserstein distance is defined as follows $$d_{W_p}(X,Y) = \Biggr(\underset{\pi \in \mathcal{J}(\mu, \nu)}{inf}\int|x-y|^p d\pi(x,y)\Biggr)^{\frac{1}{p}}$$

where $\mathcal{J}(\mu, \nu)$ denote all joint distribution $\pi$ for $(X,Y)$ that have marginals $\mu$ e $\nu$.

In this paper (arXiv link) it tells

if Y is a real-valued random variable with Lebesgue density bounded above by C > 0, then for any real-valued random variable X

$$ d_K(X,Y) \leq \sqrt{2Cd_{W_1}(X,Y)} $$ and in this paper (arXiv link) I've found that, when $Y \sim \mathcal{N}(0,1)$, $C = 2$, so $$ d_K(X,Y) \leq 2\cdot\sqrt{d_{W_1}(X,Y)} $$

My questions is:

  1. What is the value of $C$ when $Y$ is a centered non standard gaussian so $Y \sim \mathcal{N}(0,\sigma^2)$? (it should be the upper bound to the Lebesgue density of Y, but I don't know what the Lebesgue density is and how to compute it for the centered non standard gaussian distribution)
fabianod
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1 Answers1

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I don't think that the authors of the second paper claim $C=2$ (for $\mathcal{N}(0,1)$), they only state (in page 4) that $d_{K}(X,N)\leq 2\sqrt{d_{W_1}(X,N)}$ where $N\sim\mathcal{N}(0,1)$. This upper bound is actually weaker than the first upper bound that you have mentioned. Either the authors just give some upper bound that they are happy with or they are not aware of the stronger result.

The density of $\mathcal{N}(\mu, \sigma^2)$ is $p(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{\frac{-(x-\mu)^2}{2\sigma^2}}$. This function assumes its maximum value at $x=\mu$ and $p(\mu) = \frac{1}{\sqrt{2\pi\sigma^2}}$. Consequently, we have $$ d_K( X,Y) \leq \sqrt{\frac{2}{\sqrt{2\pi\sigma^2}} d_{W_1}(X,Y)} = \Big(\frac{2}{\pi}\Big)^{\frac{1}{4}}\;\frac{1}{\sqrt{\sigma}}\;\sqrt{d_{W_1}(X,Y)} $$ for $Y\sim \mathcal{N(\mu,\sigma^2)}$.

Levent
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