Parameters to describe a binary sequence? (first-time post alert)
Hello All, 
I have an empirical binary sequence (i.e. 0/1) of observations taken for equal discrete time intervals. (A typical length of such a sequence would be about 120). The probability for a "1" may decrease over time.
Now I'd like to code a procedure which generates "similar" sequences, so that I can obtain longer series with "essentially" the same behaviour; and in order to do that I'd need parameters describing the sequences. 
So the goal here is "merely" description, not "real statistics" in the sense of trying to estimate any "true" parameters from those sample sequences.
In which general direction should I be looking? Moving averages? Autocorrelation? Markov chains? (Probably not the latter: it may well be that event(i) "looks back further" than event(i-1) ) Something completely different?
I'm aware it's a rather broad (and probably ill-defined :-) question, but I am not expecting detailed analyses or guidelines; I'm asking "just" for some appropriate pointers or keywords to get me started, so that I do not take off into a completely wrong direction.
Essentially, I am trying to find the right terms for the search engines, and then will take it from there... :-)
Thanks for your help!

For some arbitrarily chosen (but not too big) $m$, consider the sequence as an order-$m$ Markov chain.  That is, you look at the conditional probabilities for $a_t$ given $a_{t-1}, a_{t-2}, \ldots, a_{t-m}$. 
