The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples, we consider random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study random determinants of random matrices of the so-called tenfold way, including the Bogoliubov–de Gennes and chiral ensembles from mesoscopic physics. We show that fixed-trace matrix ensembles can be analyzed as well. Finally, we add fine asymptotics for the [Formula: see text]-dimensional volume of the simplex with [Formula: see text] points in [Formula: see text] distributed according to special distributions, which is strongly correlated to Gram matrix ensembles. We use the framework of mod-[Formula: see text] convergence to obtain extended limit theorems, Berry–Esseen bounds, precise moderate deviations, large and moderate deviation principles as well as local limit theorems. The work is especially based on the recent work of Dal Borgo et al. [Mod-Gaussian convergence for random determinants, Ann. Henri Poincaré (2018)].

中文翻译：

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具有 Gamma 型矩的模型的精细渐近

本文的目的是为具有 Gamma 类型矩的随机变量提供精细的渐近。在这些示例中，我们考虑了具有不同维度的 Laguerre 和 Jacobi beta 集合的随机行列式（观察到的变量的数量和测量的数量不同并且可能不同）。除了经典随机矩阵模型（GOE、GUE、GSE）的戴森三重方式外，我们还研究了所谓的十重方式随机矩阵的随机行列式，包括来自介观物理学的 Bogoliubov-de Gennes 和手性系综。我们表明，也可以分析固定迹线矩阵集合。最后，我们为单纯形的 [Formula: see text] 维体积添加精细渐近线，其中 [Formula: see text] 中的 [Formula: see text] 点按照特殊分布分布，这与 Gram 矩阵集成密切相关。我们使用 mod-[Formula: see text] 收敛的框架来获得扩展极限定理、Berry-Esseen 界、精确的中等偏差、大偏差和中等偏差原理以及局部极限定理。这项工作特别基于 Dal Borgo 等人最近的工作。[随机行列式的 Mod-Gaussian 收敛，Ann。亨利·庞加莱 (2018)]。