Let [Formula: see text] be a non-atomic, infinite Radon measure on [Formula: see text], for example, [Formula: see text] where [Formula: see text]. We consider a system of freely independent particles [Formula: see text] in a bounded set [Formula: see text], where each particle [Formula: see text] has distribution [Formula: see text] on [Formula: see text] and the number of particles, [Formula: see text], is random and has Poisson distribution with parameter [Formula: see text]. If the particles were classically independent rather than freely independent, this particle system would be the restriction to [Formula: see text] of the Poisson point process on [Formula: see text] with intensity measure [Formula: see text]. In the case of free independence, this particle system is not the restriction of the free Poisson process on [Formula: see text] with intensity measure [Formula: see text]. Nevertheless, we prove that this is true in an approximative sense: if bounded sets [Formula: see text] ([Formula: see text]) are such that [Formula: see text] and [Formula: see text], then the corresponding particle system in [Formula: see text] converges (as [Formula: see text]) to the free Poisson process on [Formula: see text] with intensity measure [Formula: see text]. We also prove the following [Formula: see text]-limit: Let [Formula: see text] be a deterministic sequence of natural numbers such that [Formula: see text]. Then the system of [Formula: see text] freely independent particles in [Formula: see text] converges (as [Formula: see text]) to the free Poisson process. We finally extend these results to the case of a free Lévy white noise (in particular, a free Lévy process) without free Gaussian part.

中文翻译：

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自由独立粒子系统对自由泊松过程的逼近

令 [Formula: see text] 为 [Formula: see text] 上的非原子的无限氡测量，例如，[Formula: see text] where [Formula: see text]。我们考虑一个有界集合[公式：见文本]中自由独立粒子[公式：见文本]的系统，其中每个粒子[公式：见文本]在[公式：见文本]上有分布[公式：见文本]和粒子的数量，[公式：见文本]，是随机的，具有参数 [公式：见文本] 的泊松分布。如果粒子是经典独立的而不是自由独立的，那么这个粒子系统将是对[公式：参见文本]的泊松点过程的[公式：参见文本]的限制，具有强度测量[公式：参见文本]。在自由独立的情况下，这个粒子系统不是自由泊松过程对[公式：见文本] 与强度测量 [公式：见文本]。然而，我们证明这在近似意义上是正确的：如果有界集 [Formula: see text] ([Formula: see text]) 满足 [Formula: see text] 和 [Formula: see text]，则对应的[公式：见文本]中的粒子系统收敛（如[公式：见文本]）到具有强度测量[公式：见文本]的[公式：见文本]上的自由泊松过程。我们还证明了以下 [Formula: see text]-limit：令 [Formula: see text] 是自然数的确定性序列，使得 [Formula: see text]。然后[公式：见文]中的[公式：见文]自由独立粒子系统收敛（如[公式：见文]）到自由泊松过程。我们最终将这些结果扩展到自由 Lévy 白噪声的情况（特别是，