Epsilon-independence between two processes
We study the notion of $\varepsilon $-independence of a process on finitely (or countably) many states and that of $\varepsilon $-independence between two processes defined on the same measure preserving transformation. For that we use the language of entropy. First we demonstrate that if a process is $\varepsilon $-independent then its $\varepsilon $-independence from another process can be verified using a simplified condition. The main direction of our study is to find natural examples of $\varepsilon $-independence. In case of $\varepsilon $-independence of one process, we find an example among processes generated on the induced (first return time) transformation defined on a typical long cylinder set of any given process of positive entropy. To obtain examples of pairs of $\varepsilon $-independent processes we have to make an additional assumption on the master process. Then again, we find such pairs generated on the induced transformation as above. This is the most elaborate part of the paper. While the question whether our assumption is necessary remains open, we indicate a large class of processes where our assumption is satisfied.