Upper bound Wasserstein distance by $\chi^2$ distance

Question

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.

Answer 1

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 $f$.