If in a binomial distribution, the Bernoulli trials are independent and have different success probabilities, then it is called [Poisson Binomial Distribution](https://en.wikipedia.org/wiki/Poisson_binomial_distribution). Such a question has been previously answered [here](https://stats.stackexchange.com/questions/93852/sum-of-bernoulli-variables-with-different-success-probabilities) and [here](https://stats.stackexchange.com/questions/9510/probability-distribution-for-different-probabilities).

How can I do a similar analysis in the case of a multinomial distribution? For instance, if a $k$-sided die is thrown $n$ times and the probabilities of each side showing up changes every time instead of being fixed (as in the case of regular multinomial distribution), how can I calculate the probability mass function of such a distribution? We assume that we have access to $\{\mathbb{p\_i}\}\_1^n$ where $\mathbb{p\_i}$ is a vector of length $k$ denoted the probability of each of the $k$ sides showing up in the $i^{th}$ trial.

Note: I have asked this question on stats.stackexchange as well, but I feel it is more pertinent here.