Using cross correlation to infer dependence, can it be done? I have a very particular question, I have seen a similar one here, but my knowledge is too limited to make use of it. I will try to explain myself as clearly as possible... Wish me luck!
I have a sequence of (a priori) probabilities of a binary variable, we could say $X_i \sim Ber(p_i)$, i.e. $P(X_i = 1) = 1 - P(X_i = 0)=p_i$. I know that $X_i$ are not independent, but I don't know exactly $P(X_i|X_{j\neq i})$ nor $P(X_i|X_{j_1\neq i},...,X_{j_n\neq i})$ (I don't know anything about their dependence), and I want to know $$P(\sum_{j=-K}^{j=K} X_{i+j} =0)$$
What kind of knowledge/hypothesis do you think I need in order to approximate this more efficiently? Assuming that they are independent does not work well enough. What would you do if you found a similar situation? I am not an expert at all in this matters, and the more I learn about statistics the less I know! 
For example, I can see that the probabilities form small "triangle" shapes, so maybe something like $P(X_i = 1 | X_{i-1} = 1, X_{i-2} = 0) = P(X_i = 0 | X_{i-1} = 0, X_{i-2} = 1) $ can help? If so, is there any way to use this? As I said, I have no idea...
Thank you very much for your help!
edit: I think the title is not very good, but I don't know how to explain it better... One more question, can I say $Y_i = X_i - X_{i-1}$ and try to see $P(X_i|Y_{i-1})$?
 A: O.K., I think I found a way of doing it with the correct assumptions. Although it is only useful for my particular problem, maybe somebody can tell me if I am being too "sloppy", correct me or maybe my approach might be useful to somebody in the future.
First of all, I had not realized that the "triangles" account for "long  independent events". This means that $P(X_i = 0 | \sum_{j\neq i, |j-i| < D} X_j = 1) = 1$ in my notation ($X_i$ represents that an event starts in the moment $i$). What I started doing was a smoothing of $2D$ of $p_i$, so I took the average (can I do this?) in windows. This gave me a way of seeing how many events are there in a sequence.

The orange line is the original $p_i$ sequence, the black one is the smoothed. After this, in a window of length $2K$, I count how many peaks are there (how many possible events), and the probability of each event is the sum of probabilities from the beginning until the end of the "hill", although, as can be seen in the picture, sometimes they can overlap, but I have no idea of how can I take that into account. Then, the probability of no event happening in the window of length $2K$ is the product of probabilities of no "long event" happening in this window.
Do you think it is a good answer? Do you have any comment/suggestion? Thank you very much.
