Truncation, Sampling, and Huber Geometry for Supremum Characteristic-Function Independence Testing

We analyze supremum characteristic-function tests of mutual independence, including their behavior under sampling and robust contamination. The choice of frequency window is essential: for generic finite scalar samples of even size, an unrestricted empirical supremum attains the universal population ceiling regardless of the true dependence. A fixed window instead admits explicit uniform deviations of order \(n^{-1/2}\). For Huber contamination of two-block independence, we determine the sharp contaminated-null envelope \(b_{\rm null}(\varepsilon)=4\varepsilon(1-\varepsilon)\) when \(\varepsilon\le1/2\), and \(1\) above that level. More importantly, we identify the distinct, exact clean-law nonidentifiability threshold \(\varepsilon_c=1-1/\sqrt2\): at or above this contamination level, even maximally dependent clean laws can be observationally indistinguishable from independent ones. Under specified fixed-support and frequency assumptions, the local minimax separation rate is \(\Theta(\varepsilon+n^{-1/2})\); the complete-row replacement model has the analogous rate \(\Theta(\eta+n^{-1/2})\). The general finite-contamination clean-law overlap profile remains open. The results separate statistical consistency, robustness and geometric identifiability.