Multidimensional Free Poisson Limits on Free Stochastic Integral Algebras

In this paper, we prove four-moment theorems for multidimensional free Poisson limits on free Wigner chaos or the free Poisson algebra. We prove that, under mild technical conditions, a bi-indexed sequence of free stochastic integrals in free Wigner algebra or free Poisson algebra converges to a free sequence of free Poisson random variables if and only if the moments with order not greater than four of the sequence converge to the corresponding moments of the limit sequence of random variables. Similar four-moment theorems hold when the limit sequence is not free, but has a multidimensional free Poisson distribution with parameters $λ>0$ and $α=\{α_i: 0\ne α_i\in \mathbb{R}, i=1, 2, \cdots\}$.