Given two random variables $X,Y$ which are both $\mathbb{N}$-valued and have the same expected value (which is some fixed positive constant), and denote their probability mass functions by ${\bf p} = (p\_0,p\_1,\ldots)$ and ${\bf q} = (q\_0,q\_1,\ldots)$, respectively. We also assume that $q\_n > 0$ for all $n \in \mathbb N$. I am pretty sure one can upper bound the Wasserstein distance (of order 1) $W\_1({\bf p},{\bf q})$ by the so-called $\chi^2$ distance defined via $$\chi^2({\bf p},{\bf q}) = \sum\limits\_{n\geq 0} \frac{|p\_n-q\_n|^2}{q\_n} = \sum\limits\_{n\geq 0} \frac{p^2\_n}{q\_n} - 1.$$ However, I fail to found any specific reference containing a statement like $$W\_1({\bf p},{\bf q}) \leq f\left(\chi^2({\bf p},{\bf q})\right)$$ for some appropriate non-negative function $f$ which vanishes at the origin. I will appreciate any help in locating a suitable reference for the advertised bound...




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Remark: I forgot to mention that ${\bf p}$ and ${\bf q}$ have the same mean value in my original post.



Upper bound Wasserstein distance by $\chi^2$ distance
Such a real-valued function $f$ does not exist.


Indeed, for any natural $N$, let
$$(p\_N,p\_{2N},p\_{3N})=\tfrac18(1,4,2),\ (q\_N,q\_{2N},q\_{3N})=\tfrac18(2,2,3),$$
so that $p\_N+p\_{2N}+p\_{3N}=q\_N+q\_{2N}+q\_{3N}=\frac78$ and $Np\_N+2Np\_{2N}+3Np\_{3N}=Nq\_N+2Nq\_{2N}+3Nq\_{3N}$. Next,
for $n\in J:=\{0,1,\dots\}\setminus\{N,2N,3N\}$, let $q\_n$ be any positive real numbers such that $\sum\_{n\in J}q\_n=1-\frac78$, and let $p\_n=q\_n$ for $n\in J$.


Then $EX=EY$ and the $\chi^2$ distance between the distributions of $X$ and $Y$ is a certain positive real number, not depending on $N$.


On the other hand, the Wasserstein distance between the distributions of $X$ and $Y$ is $\sim cN\to\infty$ (as $N\to\infty$) for a certain positive real number $c$. (this easily follows from, say, the [known expression for such Wasserstein distance][1]


So, your desired inequality cannot hold for any real-valued function $f$.


