The standard way that this works is that you have 2 Banach spaces (X [say functions with 2 moments] and Y [say L^1]) that are nicely contained as subspaces of a big underlying space Z. and you have a linear operator T: X->R and T: Y->R.

Now interpolation allows you to define a scale of “in between” spaces (X,Y)_t (this is another Banach space nicely contained in Z) such that

||T||_{(X,Y)_t} \leq C ||T||_X^t ||T||_Y^{1-t},

where those norms are operator norms and 0<t<1.

So basically if T is some sort of error measure such that ||T||_Y <C and ||T||_X < n^{-k}, then

|| T ||_{(X,Y)_t} <= C n^{-tk}.

Now, I'm not likely to find the time to work it all the way through, but it would be quite surprising to me if there wasn't a way to do this so that (X,Y)_t was the space of all functions with 2t (<2) moments.

(This is especially true given that you can think of functions with k moments as weighted L^1 spaces, so Z = L^1 is a natural enveloping space)

]]>Perhaps the only exception is low-dimensional QMC, which usually lives in spaces of bounded (H-K) variation, which is not unlike the space of functions with integrable

mixed first derivatives.

So basically, while it may not be unrealistic to want the expectation of a function that is only integrable, I’m not sure we currently (or at the increment of time before these two papers) have any (sampling-based) methods for doing that. I guess I was asking if this was a “build it and they will come”-type situation or if there was a demand in the community.

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