Quantifying reduction in variance of AR(1) process after moving average smoothing

Hi there,

I wondered if anyone out there could provide some hints/suggestions to the following.

In an AR(1) process of the form

http://upload.wikimedia.org/math/7/9...6f14310af0.png

The variance can be predicted as:

http://upload.wikimedia.org/math/9/9...15ade5aec9.png

epsilon is a white noise process (normal distributed) and sigma^2 is the variance of that noise.

I would like to be able to calculate the variance of the AR(1) process after successive smoothing by a simple unweighted moving average for various window sizes. I have thus far looked at the problem analytically with real data, and as the window size of the moving average increases the variance falls, and the way in which it falls (i.e. rate and shape) is dependent on http://upload.wikimedia.org/math/3/5...953781cfdb.png. Clearly, when the moving average window is equal to N (the size of the dataset analysed) then the variance is 0.

Ultimately therefore, is there a way of determining the expected variance of the AR(1) in terms of http://upload.wikimedia.org/math/3/5...953781cfdb.png, N, and the size of the moving average window?