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Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is $$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$ if $X$ and $Y$ take values in the a metric space $(S,d)$.

Question: Does this metric have an official name?

Wikipedia calls it the "Lévy metric for $L^0$". Also, someone once told me they thought it was called the Lévy metric. I can't find it anywhere else. (If you Google™ it, don't use anything written by me as evidence that it is called the Lévy metric!) Moveover, Lévy metric is the name for another metric.

I also know this metric metricizes convergence in probability (measure), and is equivalent to the Ky Fan metric (which I have also seen called the probability distance): $$K(X,Y) = \inf\{\varepsilon \geq 0: P(d(X,Y) > \varepsilon) \leq \varepsilon\}.$$

I have also seen this very similar metric to mine $$K^*(X,Y) = \mathbb{E}\left[\frac{d(X,Y)}{1 + d(X,Y)} \right]$$ called one of the "Ky Fan metrics".

I know I could just switch to the Ky Fan metric, but I wrote a long paper using this metric $\rho$ (calling it the Lévy metric), and I don't want to have to go through the whole thing and switch to the Ky Fan metric (and change my calculations) just because I know don't know what to properly call it. (Also, I like this metric since it is reminiscent of the $L^1$-norm.)

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    $\begingroup$ I was wondering: the random variable $(X,Y)$ is measurable with respect to the product of the Borel $\sigma$-algebras ${\cal B}(S)\otimes{\cal B}(S)$. The map $d:S\times S\to\mathtt{R}^{+}$ being continuous is measurable with respect to the Borel $\sigma$-algebra ${\cal B}(S\times S)$. Don't you need to assume something like $S$ is separable to ensure that ${\cal B}(S\times S)={\cal B}(S)\otimes{\cal B}(S)$ so that $d(X,Y)$ is a measurable map and your expectation is well-defined? $\endgroup$
    – Noel Vaillant
    Commented Aug 11, 2013 at 5:59
  • $\begingroup$ @NoelVaillant, yes indeed. I always work with separable metric spaces, so I sometimes forget to mention that assumption. I added it. $\endgroup$
    – Jason Rute
    Commented Aug 11, 2013 at 14:28

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There is already a Lévy metric, or more generally the Lévy-Prokhonov metric, which is defined on distribution functions rather than random variables. However, as the following article points out, they are highly related despite this fundamental difference ("The Metrics of Prokhorov and Ky Fan for Assessing Uncertainty in Inverse Problems"). I can't say that the name Lévy metric does not sometimes refer to your metric $\rho$, but given the equivalence of $\rho$ with $K$, it would seem less confusing to just refer to it as a Ky Fan metric.

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  • $\begingroup$ I just noticed that the OP specifically mentioned the existence of another Levy metric, but I'm going to leave the answer anyway. $\endgroup$
    – Jacob Maibach
    Commented Mar 20, 2022 at 20:26

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