measure if some binary events happen in random or are correlated I have thousands of binary processes of uneven length like those below:
x0 = [(0,0), (10,0), (12,1), (20,1), (30,1), (32,0]
x1 = [(0,0), (5,0), (11,0), (27,1), (32,0), (62,1), (71,0)]
x2 = [(0,1), (3,1), (19,1), (37,0)]
x3 = [(0,0), (13,0), (18,1), (27,1), (29,1), (34,1), (38,1), (47,1), (52,0), (55,1)]

The first element in each tuple is the time and the second element is if an event happens.
I need some statistical tests to evaluate if the event (second tuple element=1) happens randomly or somehow correlates with the past observations.
Can you please give me some hints on which family of statistical methods could help?

There is an interesting angle from information theory. Randomness of data has several applications in that fiield. A totally random binary series for example cannot be compressed. (A very predictable series can be.)
One example of such a test for randomness of a binary serues can be found here Using Information Theory Approach to Randomness Testing.
One might use and AND operator on two or more series to see if a combination of these series are less random.
I would be interested in the application domain.
