We study contextual bilateral trade under full feedback when trader valuations have bounded density but infinite variance. We first extend the self-bounding property of Bachoc et al. (ICML 2025) from bounded to real-valued valuations, showing that the expected regret of any price $π$ satisfies $\mathbb{E}[g(m,V,W) - g(π,V,W)] \le L|m-π|^2$ under bounded density alone. Combining this with truncated-mean estimation, we prove that an epoch-based algorithm achieves regret $\widetilde{O}(T^{1-2β(p-1)/(βp + d(p-1))})$ when the noise has finite $p$-th moment for $p \in (1,2)$ and the market value function is $β$-Hölder, and we establish a matching $Ω(\cdot)$ lower bound via Assouad's method with a smoothed moment-matching construction. Our results characterize the exact minimax rate for this problem, interpolating between the classical nonparametric rate at $p=2$ and the trivial linear rate as $p \to 1^+$.

翻译：我们研究完全反馈下的情境双边交易问题，其中交易者估值具有有界密度但无限方差。我们首先将Bachoc等人（ICML 2025）的自约束性质从有界估值推广到实值估值，证明在仅有有界密度的条件下，任意价格$π$的期望遗憾满足$\mathbb{E}[g(m,V,W) - g(π,V,W)] \le L|m-π|^2$。结合截断均值估计方法，我们证明当噪声具有$p$阶矩（$p \in (1,2)$）且市场价值函数满足$β$-Hölder条件时，基于周期的算法可实现$\widetilde{O}(T^{1-2β(p-1)/(βp + d(p-1))})$的遗憾上界，并利用Assouad方法配合平滑矩匹配构造建立了匹配的$Ω(\cdot)$下界。我们的结果完整刻画了该问题的精确极小极大速率，其在$p=2$时的经典非参数速率与$p \to 1^+$时的平凡线性速率之间实现了连续插值。