## On Patch Stationariness II (RWLA,MCT,GT,AM Part X)

**Claim:** Â Suppose and . Â This last restriction causes freedom of choice of three functions, say the first three, and then . Â This basic construction of is the premise of all claims regarding patches I've proven here on my blog. Â Take an with , with form that will be defined in the course of the proof. Â Then provided the sum converges. Â If it does, this gives rise to the "stationary" (and it is also a patch).

**Proof:** Since we are looking for stationary , we are looking for with in direct analogy with discrete Markov chains. Â The star operator is explicitly as I've defined it previously implies . Â By the definition of , we have that . Â Expansion results in , where we have simplified to because the transformation to the y-axis does not change the integral result.

Next we have the expression

or

and derivatives

We want to obtain , so that the expression can be clearly defined. Â We use the tabular method to simplify the integration by parts.

and obtain

Lastly the steady-state "stationary" patch is exactly this , so that as we desired to show.

Notice that by hypothesis, since it is equal toÂ .

*Existence*. Â Lastly, notice that exists provided the sum converges. Â In previous exercises, I have shown that such converges in two specific examples: when derivatives of vanish (when is a finite polynomial) and when derivatives of are periodic but vanish for the periodic terms.