From Distance to Angle: One-Shot Detection Under Isotropic Multivariate Cauchy Noise

Abstract

We study one-shot detection under isotropic multivariate Cauchy noise using finite constellations, with emphasis on the geometric mechanisms governing symbol-level reliability. Under isotropic Cauchy noise, the maximum-likelihood rule induces the same Euclidean Voronoi decision regions as in the Gaussian case, so the distinction lies not in the decision geometry itself but in how probability mass is distributed over these fixed regions. In the small-noise regime, we derive a reciprocal distance-spectrum upper bound for the symbol error probability (SEP), showing that this bound, and the associated reliability descriptor, retain a longer-range dependence on the global constellation geometry than under additive white Gaussian noise. In the large-noise regime, we prove that the correct-decision probability converges to a limit determined solely by the angular measure of the associated Voronoi recession cone. These results formalize a regime-dependent transition from bound-based distance descriptors to angle-based reliability descriptors under heavy-tailed noise. Beyond asymptotic characterization, we show that these descriptors also admit a lightweight design interpretation for planar constellations under a common average power budget. The theory is further illustrated through an asymmetric four-point example exhibiting geometric collapse, a standard four-point Quadrature Amplitude Modulation (4QAM) sanity check, and finite-$\gamma$ numerical validation for both asymptotic regimes, together with descriptor-guided design comparisons that reveal collapse avoidance and reciprocal-distance burden as practically meaningful screening criteria.