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Suppose $W_r(\mu_n,\mu)\to0$, where $\mu_n$ and $\mu$ are discrete probability measures on some metric space $\Omega$, and that all measures have the same number of atoms $d$ (but not the same atoms):

$$\mu_n = \sum_{i=1}^d p_{n,i}\delta_{\theta_{n,i}}, \quad \mu = \sum_{i=1}^d p_{i}\delta_{\theta_{i}}.$$

$W_r$ here is the usual $L_r$-Wasserstein metric (a.k.a. Kantorovich–Rubinstein, optimal transport distance). Assume further that the atoms are labeled such that $\theta_{n,i}\to\theta_i$ for each $i$.

Now let $S\subset\{1,\ldots,d\}$ be any subset and consider the restriction of $\mu_n$ and $\mu$ to $S$. Let $\widetilde{\mu}_n(S)$ and $\widetilde{\mu}(S)$ denote these measures after normalization (i.e. so that each has total mass 1), i.e.

$$\widetilde{\mu}_n(S) \propto \sum_{i\in S} p_{n,i}\delta_{\theta_{n,i}}, \quad \widetilde{\mu}(S) \propto \sum_{i\in S} p_{i}\delta_{\theta_{i}}.$$

Is it true that $W_r(\widetilde{\mu}_n(S), \widetilde{\mu}(S))\to 0$ for any $S$?

Note. These can be interpreted as "conditional probabilities" where we are conditioning on the atoms in $S$.

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  • $\begingroup$ That is strangely formulated. If $\mu$ is a measure on $\Omega$, you can only restrict it to $S$ if $S\subseteq\Omega$. $\endgroup$ – Michael Greinecker Nov 20 '17 at 21:02
  • $\begingroup$ @MichaelGreinecker: Please see my edits above. There is an obvious bijection between $S$ and a subset of $\Omega$, but I realize this wasn't clear from my original post. $\endgroup$ – JohnA Nov 20 '17 at 21:03
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    $\begingroup$ The formulation is still strange. If, in the definition of $\mu_n$, you replace ${}_{n,i}$ for ${}_{n,\pi(i)}$, where $\pi$ is any permutation of the set $\{1,\dots,d\}$, then of course the measure $\mu_n$ will not change, but the measure $\tilde\mu_n$ may change quite dramatically. $\endgroup$ – Iosif Pinelis Nov 20 '17 at 21:21
  • $\begingroup$ @IosifPinelis Thanks - with the added assumption on the labeling (which is without loss of generality), this issue goes away. $\endgroup$ – JohnA Nov 20 '17 at 22:23
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The edited version of the question requires that $\langle\theta_{i,n}\rangle$ converges to $\theta_i$ for all $i$. In that case, the result is true. It suffices to show that we also have $\langle p_{i,n}\rangle$ converging to $p_i$. This is easy to see when we note that the Wasserstein-metric metrizes a topology stronger than the topology of weak convergence of measures under which $\langle\mu_n\rangle$ converges to $\mu$ if and only if $\lim_n\int g~\mathrm d\mu_n=\int g~\mathrm d\mu$ for every bounded continuous function $g$. Let $g_i$ be a bounded continuous function such that $g(\theta_i)=1$ and for each $j\neq i$ there is a neighborhood $U_j$ of $\theta_j$ such that $g$ vanishes on $U_j$. For example, we can take $g_i$ to be $\max\{1-k d(\theta_i),0\}$ for $k$ large enough. Then for $n$ large enough we have $g(\theta_{j,n})=0$ for all $j\neq i$. For such $n$, we have $\int g_i~\mathrm d\mu_n=p_{i,n}g(\theta_{i,n})$. Since $\lim_n g(\theta_{i,n})=g(\theta_i)$, we must have $\lim_np_{i,n}=p_i$.

Without the assumption that $\langle\theta_{i,n}\rangle$ converges to $\theta_i$, this fails for trivial reasons: You are not really conditioning on atoms but some arbitrary labeling of atoms. Let $\Omega=\{1,2\}$, $\theta_1=1$, $\theta_2=2$, $p_1=2/3$, $p_2=1/3$. For $n$ odd, let $\theta_{n,1}=1$, $\theta_{n,2}=2$, $p_{n,1}=2/3$, $p_{n,2}=1/3$. For $n$ even, let $\theta_{n,1}=2$, $\theta_{n,1}=2$, $p_{n,2}=2/3$, $p_{n,1}=1/3$. The sequence $\mu_n$ is constant and converges to $\mu$. But for $S=\{1\}$, the "conditional measures" are point masses switching between $1$ and $2$.

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  • $\begingroup$ Interesting - clearly we want to avoid such pathologies. Since $W_r(\mu_n,\mu)\to0$, we may assume WLOG that $\theta_{n,i}\to\theta_i$ for each $i$. I believe this sidesteps the kind of switching behaviour you describe above? $\endgroup$ – JohnA Nov 20 '17 at 21:40
  • $\begingroup$ In this case, the result is true. See my edit. $\endgroup$ – Michael Greinecker Nov 20 '17 at 22:00
  • $\begingroup$ Your argument appears to rest on the fact that $\theta_{n,i}\to\theta_i$ and $p_{n,i}\to p_i$ implies $W_r(\mu_n,\mu)\to0$. Is there a standard reference for this claim? $\endgroup$ – JohnA Nov 20 '17 at 22:08
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    $\begingroup$ @JohnA Convergence in the r-Wasserstein metric is equivalent to weak convergence plus convergence of the first r moments. This clearly holds here. The condition is mentioned in the Wikipedia-article you linked; I guess it is in one of the references. $\endgroup$ – Michael Greinecker Nov 20 '17 at 22:12

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