Conformal inference is a versatile tool for building prediction sets in regression or classification. We study the false coverage proportion (FCP) in a transductive setting with a calibration sample of $n$ points and a test sample of $m$ points. We identify the exact, distribution-free, asymptotic distribution of the FCP when both $n$ and $m$ tend to infinity. This shows in particular that FCP control can be achieved by using the well-known Kolmogorov distribution, and puts forward that the asymptotic variance is decreasing in the ratio $n/m$. We then provide a number of extensions by considering the novelty detection problem, weighted conformal inference and distribution shift between the calibration sample and the test sample. In particular, our asymptotic results allow to accurately quantify the asymptotic behavior of the errors when weighted conformal inference is used.

翻译：共形推断是构建回归或分类预测集的一种通用工具。我们研究了在转导设置下的错误覆盖比例（FCP），其中校准样本包含 $n$ 个点，测试样本包含 $m$ 个点。当 $n$ 和 $m$ 均趋于无穷大时，我们确定了 FCP 精确的、无分布的渐近分布。这特别表明，通过使用著名的柯尔莫哥洛夫分布可以实现 FCP 控制，并指出渐近方差随比值 $n/m$ 增大而减小。随后，我们通过考虑新颖性检测问题、加权共形推断以及校准样本与测试样本之间的分布偏移，提供了若干扩展。特别地，我们的渐近结果能够精确量化使用加权共形推断时误差的渐近行为。