The linear combination of Student's $t$ random variables (RVs) appears in many statistical applications. Unfortunately, the Student's $t$ distribution is not closed under convolution, thus, deriving an exact and general distribution for the linear combination of $K$ Student's $t$ RVs is infeasible, which motivates a fitting/approximation approach. Here, we focus on the scenario where the only constraint is that the number of degrees of freedom of each $t-$RV is greater than two. Notice that since the odd moments/cumulants of the Student's $t$ distribution are zero, and the even moments/cumulants do not exist when their order is greater than the number of degrees of freedom, it becomes impossible to use conventional approaches based on moments/cumulants of order one or higher than two. To circumvent this issue, herein we propose fitting such a distribution to that of a scaled Student's $t$ RV by exploiting the second moment together with either the first absolute moment or the characteristic function (CF). For the fitting based on the absolute moment, we depart from the case of the linear combination of $K= 2$ Student's $t$ RVs and then generalize to $K\ge 2$ through a simple iterative procedure. Meanwhile, the CF-based fitting is direct, but its accuracy (measured in terms of the Bhattacharyya distance metric) depends on the CF parameter configuration, for which we propose a simple but accurate approach. We numerically show that the CF-based fitting usually outperforms the absolute moment -based fitting and that both the scale and number of degrees of freedom of the fitting distribution increase almost linearly with $K$.

翻译：在众多统计应用中，经常出现学生t随机变量（RVs）的线性组合。然而，学生t分布对卷积运算不封闭，因此无法推导出K个学生t随机变量线性组合的确切通用分布，这促使我们采用拟合/近似方法。本文重点关注每个t变量的自由度均大于2的情形。注意到学生t分布的奇数阶矩/累积量为零，且当矩/累积量的阶数超过自由度时，偶数阶矩/累积量不存在，这使得基于一阶或二阶以上矩/累积量的传统方法无法适用。为解决该问题，本文利用二阶矩结合一阶绝对矩或特征函数（CF），提出将此类分布拟合为缩放后的学生t随机变量分布的方法。基于绝对矩的拟合从K=2个学生t随机变量线性组合的情况出发，通过简单迭代过程推广至K≥2的情形。而基于特征函数的拟合更为直接，但其精度（以Bhattacharyya距离度量）取决于特征函数参数配置，对此我们提出一种简单而精确的方法。数值结果表明，基于特征函数的拟合通常优于基于绝对矩的拟合，且拟合分布的尺度参数和自由度均随K近似线性增长。