Disclaimer: First, the usage of "coupling" in the title is not of the usual definition in probability theory. Second, cross-posting from stats.stackexchange.com.
Suppose I have a set of random variables $\{X_1, X_2, \dots, X_n\}$, indexed by time $t$, and where $X_i(t) \in \{0, 1\}$, $\forall i\in \{1, \dots, n\}, \forall t \in \mathbb{Z}^+$.
I want to model a process as follows: at each time $t$, the value of $X_i(t+1)$ is a random function of the values of the set of all other random variables, i.e. $\{X_1, \dots, X_n\} \backslash X_i$. The random function doesn't matter too much, but one example could be as follows:
- Sample $z$ random elements from $\{X_1, \dots, X_n\} \backslash X_i$, and place them into set $S$ (i.e. sample $z$ r.v. from the set of all random variables not including $X_i$).
- Create new empty set $S'$. For every $X_j(t) \in S$, add 0 to $S'$ if $X_j(t)$ is 0, otherwise add 1 with probability 1/2 each if $X_j(t)$ is 1.
- If the sum all all elements in $S$ is less than $z/2$, then set $X_i(t+1)$ to be 0. Otherwise set it to 1.
One way to model this process is to set up a Markovian analysis, where each state is a 0-1 vector of length $n$, for a total of $2^n$ states. If you start adding other parameters, these types of Markovian constructions quickly blow up.
Is there another another general way to analyze these types of random fields of many random variables, whose values depends on the values of other random variables and update over time?
Any links to closely related literature would be immensely appreciated!