I'm far from being a mathematician myself, but I will give my opinion on this. Independence (probability theory) can mean that two events are independent when the occurrence of one does not affect the probability of the other.
Let us say that when price goes up after a unit of time, it can be considered an event. Now let's say that price going down after a unit of time is an event. From that, we have two events independent of each other.
If we were to count how many times in a row that the price going up event occurred, then compare its count with how much Time occurred, we could possibly find some patterns in there somewhere. These patterns become independent for the price going up event. The same can be done for the price going down event, and it will have its own independent patterns.
Now to determine that a pattern itself is independent, we need to use as much data as available (true accuracy would come from having ALL of the data), then we compute to find a pattern that has only occurred once.
As an example, every time the price goes up, it becomes the new highest occurrence of the amount of times it has ever went up. So that should mean that it will stay an independent pattern until the same pattern completely occurs again. Meaning, every time the price goes up, the pattern that reaches half of the total occurrences of going up will finally stop being independent.
However, the more skewed and different variables that we add to a pattern, the more likely we are to finding new independent patterns. For the things I'm working on, I am more interested in using creative patterns (that occur a lot) that are made out of multiple events, then compare history to find events that seem to follow(happen in the future) as a pattern.
As far as to prove or make a Proof of anything, I'm not ready for that challenge at the moment.
