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I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$ with finite second moments, and $W_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question is: (1) As $\mu$ and $\nu$ are distinct, hence $W_2(\mu,\nu)\neq 0$. Is it possible that there is a sequence of distributions $\mu_i, i\geq 1$ on $\mathbb{R}^n$ such that $\lim_{i\to\infty} W_2(\mu*\mu_i,\nu*\mu_i)=0$? Here $*$ is the convolution.

While I was trying to figure out the answer using Prokhorov's theorem, I realized that I need to answer the following more elementary looking question (though I do not have an answer yet):

(2) If $\mu$ and $\nu$ are two distinct probability measures on $\mathbb{R}^n$, then is it possible that there is another probability measure $\mu'$ such that $\mu*\mu' = \nu*\mu'$?

Some special cases can be easily dealt with (to give a negative answer), using for example Fourier transform. However, I am unsure about the general case.

Any comments and references to these two problems are highly appreciated!

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The answer is yes to the second question, and hence yes to the first question as well.

Indeed, it is easy to check that the functions $f$ and $g$ given by $$f(t):=\max(0,1-|t|)$$ and $$g(t):=\sum_{k=-\infty}^\infty f(t-2k)$$ for real $t$ are characteristic functions: $$f(t)=\int_{-\infty}^\infty e^{itx}\mu(dx)$$ and $$g(t)=\int_{-\infty}^\infty e^{itx}\nu(dx),$$ where $$\mu(B):=\int_B dx\,\frac{1-\cos x}{\pi x^2}$$ and $$\nu(B):=\frac{1(0\in B)}2 +\frac2{\pi^2}\,\sum_{k=-\infty}^\infty \frac{1((2k-1)\pi\in B)}{(2k-1)^2} $$ for all Borel sets $B\subseteq\mathbb R$.

We also have $$gf=f^2=ff.$$

So, if $\mu':=\mu$, then we have $\mu\ne\nu$ but $\mu*\mu'=\nu*\mu'$, as claimed.

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  • $\begingroup$ Nice example. Thanks. $\endgroup$
    – F J
    Apr 7 at 13:50

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