In this paper, we focus on the asymptotic behavior of a recently popular risk measure called the tail moment (TM), which has been extensively applied in the field of risk theory. We conduct the study under the framework in which the individual risks of a financial or insurance system follow convolution equivalent or Gamma-like distributions. Precise asymptotic results are obtained for the TM when the individual risks are mutually independent or have a dependence structure of the Farlie-Gumbel-Morgenstern type. Moreover, based on some specific scenarios, we give an asymptotic analysis on the relative errors between our asymptotic results and the corresponding exact values. Since the model settings in this paper are not covered by traditional ones, our work fills in some gaps of the asymptotic study of the TM for light-tailed risks.

在本文中，我们关注最近流行的一种称为尾矩（TM）的风险度量的渐近行为，该度量已广泛应用于风险理论领域。我们在金融或保险系统的个体风险遵循卷积等价分布或类伽玛分布的框架下进行研究。当各个风险相互独立或具有 Farlie-Gumbel-Morgenstern 类型的依赖结构时，TM 会获得精确的渐近结果。此外，基于一些具体场景，我们对渐近结果与相应精确值之间的相对误差进行了渐近分析。由于本文的模型设置没有涵盖传统的模型设置，因此我们的工作填补了轻尾风险TM渐近研究的一些空白。