Re: Is there an analog of the central limit theorem for integrals?

_Asymptotic Theory for Econometricians_ by Halbert White
has an excellent and extended discussion of these ideas.

If memory serves, the book by Spanos,
"Probability Theory and Statistical Inference" might also
have a discussion of these topics, at a somewhat
lower level than White.

Thanks very much. Unfortunately, the library at the lab where I work (LANL) has neither of those books. It does, however, have lots and lots of others, so a third (or fourth) recommendation would be appreciated.

A development: I have found, from searching around online for things like "strong mixing" and "weak dependence" that there is a great literature on the asymptotic distributions of sums of dependent random variables. I find that the integral in the opening post can be reduced to such a sum. Look:

Recall, we are considering an integral of the form (forget the prefactor):

I(x) = int_0^x dy r(y),

where <r(y)> = 0, and <r(y1)r(y2)> = C(|y1-y2|). Without loss of generality, let's say that C(0)=1. We can make this integral into a sum as follows: Let "d" be the scale length over which C(y) falls to zero as |y|->infinity. Furthermore, let x = nd for integer n>0. Then we can write the integral as

I(x) = S_n = \sum_{m=0}^{n-1} z_m,

where

z_m = \int_{md}^{(m+1)d} dy r(y).

It is then clear that <z_m>=0, and that the z_m form a stationary sequence. Furthermore, if we assume that C(y) is positive semidefinite, and that it is monotonically decreasing in |y| (both of which are clear from the original physical context of the problem), it is easy to show that <z_m z_0> falls of like C(md). So, assuming that C(y) falls of rapidly, e.g. polynomial times an exponential, we have in S_n a sum of a stationary sequence of mean-0 random variables, with rapid fall-off of correlations along the sequence.

My gut tells me that S_n should be normally distributed asymptotically as n->infinity. In the literature, there are a great many theorems (such as those concerning "m-dependence") that just barely fail to address this particular problem. For example, this sequence is not strictly m-dependent. But so many of these theorems come so close to stating the result I want that it seems clear to me that somewhere out there the necessary theorem is floating, waiting for me to find it.

For me, this is one of those little sub-problems that hold up the progress of a much greater work (in this case a new result in shock physics), and which you beat your head against for days/weeks, to no avail. You know the ones.

Any further help would be greatly appreciated! Meanwhile, I'll keep looking. If I find the necessary result, I'll post it.

Thanks.
.