I was wondering if any arbitrary discrete probability distribution that takes nonnegative integer values and with bounded support, can be written as a summation of dependent Bernoulli variables.
For example, the Binomial distribution can be written as a summation of independent Bernoulli variables. Can the same be said, for dependent variables, for any arbitrary distribution with these characteristics?
UPDATE: Based on comments, one way to do so is by writing the r.v. $Z=\sum_{k=1}^{\infty} I(Z\leq k)$. Is there any alternative representation that kind of minimizes the interdependencies between variables? For example this makes all Bernoullis dependent on each other. But in the case of a Binomial distribution, we can surely describe it in terms of independent bernoullis