Convex combinations of i.i.d. random variables without a finite mean can behave in a strikingly different way from the finite-mean case: as the weight vector becomes more balanced, the resulting combination may become stochastically larger, rather than less dispersed. Existing results establish stochastic dominance between pairs of linear combinations-or between a convex combination and the underlying variable-under shape restrictions on the distribution and structural assumptions on the weights. We expand the class for which the general result can be derived. Nonetheless, two practical limitations remain: (i) the sufficient conditions vary across results, and (ii) being non-necessary, they exclude many relevant configurations. Moreover, under a statistical perspective, where the true distribution of the data is assumed to be unknown, these conditions cannot be checked. Motivated by this gap, we develop nonparametric procedures to test whether two linear combinations are stochastically ordered. We propose two complementary approaches: a least-favorable calibration and a bootstrap-based method.We show that both tests control size asymptotically under the null of stochastic dominance and are consistent against alternatives of non-dominance. Monte Carlo experiments illustrate the finite-sample performance of the proposed procedures across a range of models and weight configurations.

翻译：对于不存在有限均值的独立同分布随机变量的凸组合，其行为可能与有限均值情形存在显著差异：随着权重向量趋于均衡，所得组合可能随机增大，而非离散程度降低。现有研究在分布形态约束和权重结构假设下，建立了线性组合对之间——或凸组合与基础变量之间——的随机占优关系。我们扩展了可推导一般结果的分布类别。然而，仍存在两个实际局限：（i）充分条件因结果而异；（ii）这些非必要条件排除了许多相关配置。此外，在假定数据真实分布未知的统计视角下，这些条件无法被验证。受此缺口驱动，我们开发了非参数程序来检验两个线性组合是否具有随机序关系。我们提出两种互补方法：最不利校准法和基于Bootstrap的方法。我们证明两种检验在随机占优的原假设下能渐近控制检验水平，并在非占优备择假设下具有相合性。蒙特卡洛实验展示了所提方法在多种模型和权重配置下的有限样本表现。