Let be a mixing system, with .

The auto-correlation function is defined by .

In the following we assume converges. Under some extra condition, we have the central limit theorem converges to a normal distribution.

The power spectrum of is defined by (when the limit exist)

.

Note that whenever converges.

Proof. Let . Then . So

. Then we have

since .

More generally, we have

where . So exists whenever converges.

This is the power spectrum of . Some observations:

**Proposition.** Assume .

Then is well-defined, continuous function on . Moreover,

– is if for all ;

– is if decay rapidly;

– is if decay exponentially.

Some preparations.

Hardy: Let be a sequence of real numbers, such that

– for all ;

– .

Then also converges to .

Proof. Let be given, large such that for all .

Then for any , we have

;

.

Note that

Then So we can pick , which leads to

for all .

Let be differentiable, and . Let and be the Fourier series. Then .

Proof: integrate by parts.

Let be differentiable, and . Then converges.

In particular, converge for all .

Proof. Note that