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Consider a probability space $(\Omega, \mathcal{F}, P)$, as well as two sub-$\sigma$-fields $\mathcal{A}$ and $\mathcal{B}$. The Hausdorff pseudo-distance between $\mathcal{A}$ and $\mathcal{B}$ is defined as follows: $$D(\mathcal{A},\mathcal{B}):=\max\{\sup_{A\in \mathcal{A}}\inf_{B\in\mathcal{B}}P(A\Delta B) ; \sup_{B\in \mathcal{B}}\inf_{A\in\mathcal{A}}P(A\Delta B)\}, $$ where $\Delta$ stands for the usual symmetric difference between sets, i.e.: $A\Delta B = (A-B\cap A)\cup (B-B\cap A)$. Now suppose that $F, N$ are two independent random vectors of some common dimension $d\geq 1$ (we may even assume that $N$ is Gaussian and that $F$ has a density), and also that $\mathcal{A} = \sigma(F)$ and $\mathcal{B} = \mathcal{B}_{\epsilon} = \sigma(F+\epsilon N)$. Is there some known method in order to obtain estimates on the quantity $D(\mathcal{A},\mathcal{B}_\epsilon)$, as $\epsilon $ converges to zero? I am expecting that such a quantity converges to zero, and I would like to have an upper bound on this convergence (the faster the better!).

Relevant references on the topic (among many others) are a paper by Rogge


and one by van Zandt


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If $F$ degenerate and $N$ nondegenerate there is no convergence, since $\sigma(F)$ is trivial and $\sigma(F + \epsilon N) = \sigma(N)$ is not. –  Dan Feb 19 '13 at 13:25
Dan: this is a pertinent remark (even if I specified that $F$ has a density)! Anyway, I have really a poor understanding of the Hausdorff distance, whenever it is applied to arbitrary $\sigma$-fields; moreover, it seems to me (hopefully, I am wrong) that there are few examples in the literature with some explicit estimates. It seems to me that the most explicit results are in a martingale framework. I wonder whether one can build a counterexample (that is, an example where one has not convergence to zero) involving a random variable $F$ which is not trivial, say with a density. –  user31542 Feb 20 '13 at 10:17
This paper, for instance, projecteuclid.org/DPubS/Repository/1.0/… has a lot of explicit computations –  user31542 Feb 20 '13 at 10:18

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