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Question: If we have a sequence of $\mathbb{R}$-valued random variables $X_n$ which converge in the metric space of random variables defined by the energy distance to the $\mathbb{R}$-valued random variable $X$, i.e. $$D(X_n,X) \to 0$$ then is this equivalent to the sequence $X_n$ converging to $X$ in $L^2$, i.e. $$\mathbb{E}[(X_n - X)^2] \to 0?$$

My thinking is that $L^2$ convergence implies that $$\E X_n^2 - 2\E X_n X + \E X^2 \to 0$$ which looks similar to the formula of covariance, and thus makes me think that there might be a relationship with Brownian covariance and distance covariance.

Also energy distance is equal up to a constant factor to the Cramer distance (the formula for which looks like the definition of $L^2$ convergence) for real-valued random variables. This is fortuitous because I am interested primarily in the real-valued case.

Finally there seems like there might be a relationship between energy (in physics) going to zero and $L^2$ convergence, so if there is an analogous relationship between energy distance and $L^2$ convergence, that would explain why energy distance is called energy distance (Wikipedia doesn't explain the etymology of the name). I have asked questions about this relationship before on StackExchange here, here, and here.

Also it would be interesting too because then one could interpret least-squares regression as "energy-minimizing" regression. But my main goal is to argue that the $L^2$ convergence that one has in the definition of stochastic integrals (as opposed to more conventional almost sure convergence) is natural in the sense that it is equivalent pointwise convergence of the "energy difference" between the approximating processes and the stochastic integral to zero for each moment in time.

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Connections/Dominance between distances is not that straightforward to evaluate. In particular, there does not seem to be an obvious connection between the energy distance and the $L^2$ distance. See:

On choosing and bounding probability metrics.

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    $\begingroup$ @William Google is your friend: math.hmc.edu/~su/papers.dir/metrics.pdf. Regarding the Energy distance, this is a very young distance: en.wikipedia.org/wiki/Energy_distance, so, a lot of opportunity for you to do some research. $\endgroup$ – Luck Aug 19 '16 at 1:16
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    $\begingroup$ This paper is a really good introduction to all of these topics that Wikipedia just lists but doesn't explain -- thank you for sending me to it! $\endgroup$ – Chill2Macht Aug 19 '16 at 1:28

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