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.
This has been answered in a recent article entitled The Poisson Multinomial Distribution and Its Applications in Voting Theory, Ecological Inference, and Machine Learning by Zhengzhi Lin, Yueyao Wang, and Yili Hong (2022).
They describe solutions using an FFT (exact, full PMF), Monte-Carlo simulations (single point from the full PMF) or a Normal approximation.
R code is available here.