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

In article <13103647.1148664947919.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
John L. Barber <jlbarber@xxxxxxxx> wrote:
(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.

There certainly is such an analog, and even more, to sums
of dependent random variables. The condition for asymptotic
normality depends on what are called "strong mixing"
conditions, and there is much literature on this.


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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