In this paper, we study compound bi-free Poisson distributions for two-faced families of random variables. We prove a Poisson limit theorem for compound bi-free Poisson distributions. Furthermore, a bi-free infinitely divisible distribution for a two-faced family of self-adjoint random variables can be realized as the limit of a sequence of compound bi-free Poisson distributions of two-faced families of self-adjoint random variables. If a compound bi-free Poisson distribution is determined by a positive number and the distribution of a two-faced family of finitely many random variables, which has an almost sure random matrix model, and the left random variables commute with the right random variables in the two-faced family, then we can construct a random bi-matrix model for the compound bi-free Poisson distribution. If a compound bi-free Poisson distribution is determined by a positive number and the distribution of a commutative pair of random variables, we can construct an asymptotic bi-matrix model with entries of creation and annihilation operators for the compound bi-free Poisson distribution.

中文翻译：

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复合双自由泊松分布

在本文中，我们研究了双面随机变量族的复合双自由泊松分布。我们证明了复合双自由泊松分布的泊松极限定理。此外，自伴随随机变量的双面族的双自由无限可分分布可以实现为自伴随随机变量的双面族的复合双自由泊松分布序列的极限。如果复合双自由泊松分布由正数和有限多个随机变量的双面族的分布确定，该族具有几乎确定的随机矩阵模型，并且左随机变量与右随机变量在面族，则我们可以为复合双自由泊松分布构造一个随机双矩阵模型。