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

(I'm reposting this in sci.stat.math, since this seems like a better place than sci.math, where I originally posted it.)

OK, so suppose we have a random variable y which is a sum of n random variables:

y = ( \sum_{i=1}^n x_i )/n^(1/2)

Here the x_i are independent, identically distributed random variables. Without loss of generality, let's say that <x_i>=0, and <x_i^2>=1. Then the Central Limit Theorem (CLT) states that

y -> Z as n->infinity,

where Z is a standard Gaussian random variable with mean 0 and variance 1.

Now, consider a related problem. Define

y(x) = ( \int_0^x du r(u) )/x^(1/2)

Here r(u) is real-valued, stationary, stochastic process. Without loss of generality, assume that <r(u)>=0, and <r(u)^2>=1. Furthermore, assume that r(u) satisfies

<r(u1)r(u2)> = C(u1 - u2),

where the autocorrelation function C(u) is even in its argument, and falls off to zero as u->infinity, with some scale length "d". Note that C(0)=1, by construction. An example of such a C(u) could be C(u) = exp(-u/d).

Now, my question is this: Is anyone aware of any analog of the CLT for the integral y(x)? In other words, is it true that

y(x) -> SD[y(x)]*Z, as x/d -> infinity ?

Here "SD[y(x)]" means the standard deviation of the integral y(x), and Z is another standard normal random variable.

Thanks very much for any help!
.

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