General Autocorrelation Function.

An example of a class of statistics of the form (9.1) where $\{{\bf p}_m\}$ and $\{q_m\}$ are all zero are ACF estimates

$$r_t = \frac{1}{N}\; \sum_{i=t+1}^N \; x_i \; x_{i-t}, \;\;\; 0\leq t \leq N-1.$$ (9.7)

Suppose we are interested only in a selected set of ACF samples at delays $t_1, t_2 \ldots t_M$. The problem is to obtain the joint PDF of the feature vector

$${\bf z}= [r_{t_1}\; r_{t_2} \ldots r_{t_M}]^\prime,$$

denoted by

$$p(r_{t_1}, r_{t_2} \ldots r_{t_M} ; H_0).$$

The elements of ${\bf z}$ can be written as quadratic forms

$$z_m = {\bf x}^\prime {\bf P}_{t_m} {\bf x}, \;\;\; 1\leq m \leq M,$$

where

$$\begin{array}{r} {\bf P}_0 = \frac{1}{N}\;\left[ \begin{array}{c... ...\cdots \\ \vdots & \vdots & \vdots & \vdots \end{array}\right], \end{array}$$

and so forth. The pattern is such that ${\bf P}_k$ is nonzero only on the super- and sub-diagonals spaced $k$ away from the main diagonal.

If the sample mean is subtracted from ${\bf x}$ prior to calculation of the ACF estimates, the quadratic forms (9.1) still hold, but the elements of $\{{\bf P}_k\}$ are changed. For example, the $j,k$-th element of ${\bf P}_0$ is now $\delta_{jk} - 1/N$ instead of $\delta_{jk}$, where $\delta_{jk}$ is the Kronecker delta; the remaining matrices $\{{\bf P}_k\}$ are more complicated, but each element in the matrices can be evaluated by means of a single sum.