An inequality involving the Wasserstein distance and chi-squared distance

Question

$\newcommand{\N}{\mathbb N}$Let $P$ be the set of all probability mass functions on $\N_0:=\{0\}\cup\N$, where $\N:=\{1,2,\dots\}$. Let $P_{>0}$ denote the set of all $q=(q_0,q_1,\dots)\in P$ such that $q_n>0$ for all $n\in\N_0$.

A previous MO post asked the following question:

Is there a function $f\colon[0,\infty)\to[0,\infty)$ with $f(0)=0$ such that for all $p=(p_0,p_1,\dots)\in P$ and all $q=(q_0,q_1,\dots)\in P_{>0}$ we have \begin{equation} W_1(p,q)\le f(\chi^2(p,q)), \end{equation} where $W_1(p,q)$ is the Wasserstein distance of order $1$ between $p$ and $q$, and \begin{equation} \chi^2(p,q):=\sum_{n\in\N_0}\frac{(p_n-q_n)^2}{q_n}, \end{equation} the chi-squared "distance" between $p$ and $q$?

The answer to this question is negative, even with the later added condition that the means for $p$ and $q$ are the same.

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However, as will be shown below, the answer becomes positive if $f$ is allowed to depend on $q$ or, more specifically, just on the second moment \begin{equation} m_2(q):=\sum_{n\in\N}n^2 q_n \end{equation} of $q$.

Answer 1

$\newcommand{\N}{\mathbb N}$By the known expression for the Wasserstein distance, \begin{equation*} \begin{aligned} W_1(p,q)&=\int_0^\infty dx\,\Big|\sum_{n>x}(p_n-q_n)\Big| \\ &\le\int_0^\infty dx\,\sum_{n>x}|p_n-q_n| \\ &=\sum_{j\in\N_0}\int_{[j,j+1)} dx\,\sum_{n\ge j+1}|p_n-q_n| \\ &=\sum_{j\in\N_0}\sum_{n\ge j+1}|p_n-q_n| \\ &=\sum_{n\ge1}|p_n-q_n|\sum_{j=0}^{n-1}1 \\ &=\sum_{n\ge1}|p_n-q_n|\,n \\ &=\sum_{n\ge1}n\sqrt{q_n}\,\frac{|p_n-q_n|}{\sqrt{q_n}} \\ &\le\sqrt{\sum_{n\ge1}n^2 q_n}\,\sqrt{\sum_{n\ge1}\frac{(p_n-q_n)^2}{q_n}}, \end{aligned} \tag{10}\label{10} \end{equation*} by the Cauchy--Schwarz inequality. So, \begin{equation*} W_1(p,q)\le\sqrt{m_2(q)}\sqrt{\chi^2(p,q)}. \tag{20}\label{20} \end{equation*}

Moreover, taking any $q\in P_{>0}$, any $N\in\N$, any $h\in\big(0,q_0/\sum_{n=1}^N nq_n\big)$, letting \begin{equation*} p_n:=q_n+hnq_n\,1(n\le N) \end{equation*} for $n\in\N$, with \begin{equation*} p_0:=q_0-h\sum_{n=1}^N nq_n, \end{equation*} and looking back at \eqref{10}, we get \begin{equation*} W_1(p,q)=\sqrt{m_{2,N}(q)}\sqrt{\chi^2(p,q)}, \end{equation*} where $m_{2,N}(q):=\sum_{n=1}^Nn^2 q_n$. Letting now $N\to\infty$, we see that the upper bound on $W_1(p,q)$ in \eqref{20} is sharp, for each $q\in P_{>0}$.