Consider the task of generating samples from a tilted distribution of a random vector whose underlying distribution is unknown, but samples from it are available. This finds applications in fields such as finance and climate science, and in rare event simulation. In this article, we discuss the asymptotic efficiency of a self-normalized importance sampler of the tilted distribution. We provide a sharp characterization of its accuracy, given the number of samples and the degree of tilt. Our findings reveal a surprising dichotomy: while the number of samples needed to accurately tilt a bounded random vector increases polynomially in the tilt amount, it increases at a super polynomial rate for unbounded distributions.

翻译：考虑从随机向量的倾斜分布中生成样本的任务，其中该随机向量的基础分布未知，但可获得其样本。这在金融、气候科学以及罕见事件模拟等领域有应用。在本文中，我们讨论了倾斜分布的自归一化重要性采样器的渐近效率。给定样本数量和倾斜程度，我们对其精度给出了一个尖锐的刻画。我们的研究结果揭示了一个惊人的二分现象：虽然准确倾斜一个有界随机向量所需的样本数量随倾斜量呈多项式增长，但对于无界分布，其增长速率却是超多项式的。