### General Autocorrelation Function.

An example of a class of statistics of the form (8.1) where and are all zero are ACF estimates

 (8.7)

Suppose we are interested only in a selected set of ACF samples at delays . The problem is to obtain the joint PDF of the feature vector

denoted by

The elements of can be written as quadratic forms

where

and so forth. The pattern is such that is nonzero only on the super- and sub-diagonals spaced away from the main diagonal.

If the sample mean is subtracted from prior to calculation of the ACF estimates, the quadratic forms (8.1) still hold, but the elements of are changed. For example, the -th element of is now instead of , where is the Kronecker delta; the remaining matrices are more complicated, but each element in the matrices can be evaluated by means of a single sum.

Baggenstoss 2017-05-19