A 1/R Law for Kurtosis Contrast in Balanced Mixtures

Kurtosis-based Independent Component Analysis (ICA) weakens in wide, balanced mixtures. We prove a sharp redundancy law: for a standardized projection with effective width $R_{\mathrm{e}\mathrm{f}\mathrm{f}}$ (participation ratio), the population excess kurtosis obeys $\mathrm{\mid }\kappa (y)\mathrm{\mid }=O({\kappa }_{\max }\mathrm{/}{R}_{\mathrm{e}\mathrm{f}\mathrm{f}})$, yielding the order-tight $O(c_b{\kappa }_{\max }\mathrm{/}R)$ under balance (typically $c_b=O(\log R)$). As an impossibility screen, under standard finite-moment conditions for sample kurtosis estimation, surpassing the $O(1\mathrm{/}\sqrt{T})$ estimation scale requires $R\lesssim {\kappa }_{\max }\sqrt{T}$. We also show that \emph{purification} – selecting $m\text{\hspace{0.17em}⁣}\ll \text{\hspace{0.17em}⁣}R$ sign-consistent sources – restores $R$-independent contrast $\mathrm{\Omega }(1\mathrm{/}m)$, with a simple data-driven heuristic. Synthetic experiments validate the predicted decay, the $\sqrt{T}$ crossover, and contrast recovery.