**Setup:**


I have a sequence of stationary ergodic random variables $(\epsilon\_t)\_{t\in\mathbb{Z}}$ and a function $\phi:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$. Define the sequence of random functions $(\phi\_t(x) = \phi(x,\epsilon\_t))\_{t\in\mathbb{Z}}$.


Now suppose there exists a stationary ergodic sequence of random variables $(X\_t)\_{t\in\mathbb{Z}}$ such that


1. $X\_{t+1} = \phi\_t(X\_t)$ for all $t\in\mathbb{Z}$.
2. For any random variable $Y$ there exists a $\rho>1$ such that almost surely
$$
\rho^t\left|X\_{t+1} - \phi\_t \circ\phi\_{t-1}\circ\ldots\phi\_0(Y)\right| \stackrel{t\rightarrow\infty}{\rightarrow} 0.
$$


**Question:**


The second property is called invertibility and can be interpreted as the process forgetting about its past. Does anyone know about the relation of this condition with mixing, explicitly whether it implies


1. Mixing in the ergodic sense
2. $\alpha$-mixing


Thank you in advance!



Relation between invertibility and strong mixing of a time series
I don't think your condition implies anything much. Let $(\epsilon\_t)$ be your favourite ergodic 0--1 valued process; and let $X\_t=\sum\_{j=1}^\infty 2^{-j}\epsilon\_j$. Then the function $\phi$ is $\phi(x,y)=(x+y)/2$. I believe this satisfies all of your conditions for any $\rho<2$.


