Bilateral Trade Under Heavy-Tailed Valuations: Minimax Regret with Infinite Variance

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 $\pi$ satisfies $\mathbb{E}[g(m,V,W)-g(\pi ,V,W)]\le L\mathrm{\mid }m-\pi {\mathrm{\mid }}^2$ under bounded density alone. Combining this with truncated-mean estimation, we prove that an epoch-based algorithm achieves regret $\tilde{O}(T^{1-2\beta (p-1)\mathrm{/}(\beta p+d(p-1))})$ when the noise has finite $p$-th moment for $p\in (1,2)$ and the market value function is $\beta$-Hölder, and we establish a matching $\mathrm{\Omega }(\cdot )$ 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 $p\to 1^{+}$.