Generalizing Poisson's binomial distribution to the multinomial case. If in a binomial distribution, the Bernoulli trials are independent and have different success probabilities, then it is called Poisson Binomial Distribution. Such a question has been previously answered here and here.
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.

I have come across very few resources on the topic. Such a problem is called Generalized Poisson's distribution or GPB. I am listing a few of them here to help others.


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*An old paper describing the calculation of its approximate pdf.

*A 2018 paper describing an algorithm to compute its PDF using Fourier Transforms.

*An R package implementing the same.

