# Multidimensional Free Poisson Limits on Free Stochastic Integral Algebras

## TL;DR

This paper establishes four-moment theorems for multidimensional free Poisson limits within free stochastic integral algebras, characterizing convergence to free Poisson distributions through moment conditions.

## Contribution

It extends four-moment theorems to multidimensional free Poisson limits in free Wigner and free Poisson algebras, including non-free cases with specified parameters.

## Key findings

- Convergence characterized by moments up to order four.
- Results apply to free and non-free multidimensional free Poisson limits.
- Provides conditions under which stochastic integrals converge to free Poisson distributions.

## Abstract

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 $\lambda>0$ and $\alpha=\{\alpha_i: 0\ne \alpha_i\in \mathbb{R}, i=1, 2, \cdots\}$.