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Suppose $(P_n)_{n\ge 1}$ is a sequence of probability measures on a metric space $E$. Everybody knows what weak convergence of $P_n$ to a measure $P$ is.

Instead, let $(Q_n)_{n\ge 1}$ be another sequence of probability measures, and I want to ask the following question: when are $Q_n$ and $P_n$ asymptotically close in the sense of weak convergence? Of course, I can say that it means that $d(P_n,Q_n)\to 0$ where $d$ is one of the metrics compatible with weak convergence of measures. However, is there a useful theory helping to prove $d(P_n,Q_n)\to 0$? For the usual case i.e. weak convergence $P_n\to P$, there is a number of technologies based on tightness, and here we may have a situation where none of the two sequences is tight, but they still "converge to each other".

This could be useful if one of the two sequences is much simpler or better understood than the other one, and I am mostly interested in situations where the metric space $E$ is a functional space like $C$ or $D$.

UPD. It turns out that "merging" is the keyword and there is some literature on the issue (thanks to Mark Meckes for pointing to one of these papers), but it looks like there is still no tool to prove merging in functional spaces.

It seems that the most recent paper related to this topic is

Davydov & Rotar: On asymptotic proximity of distributions. J. Theoret. Probab. 22 (2009), no. 1, 82--98.

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2 Answers 2

D'Aristotile, Diaconis, and Freedman called this "merging" of probability measures, and studied relationships among different notions of it in this paper.

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This looks interesting, thanks for pointing to this paper! I just requested a copy from my library. –  Yuri Bakhtin Apr 6 '10 at 13:10

I think that 'tightness' makes sense for signed measures, with no important differences. In this case, $Q_{n} - P_{n}$ is a signed measure, and you can talk about tightness for this difference, and whether or not it converges to 0 (or something else).

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Q_n-P_n looks more appropriate for total variation convergence. The question is: is there a contentful and well-developed theory helping to prove asymptotic merging of measures. Say, the reference given by Mark Meckes provides several nonequivalent definitions, although it is not yet clear to me how far the theory is pushed in that paper. –  Yuri Bakhtin Apr 6 '10 at 13:19

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