Statistical Estimation of Extremal Characteristic-Function Functionals: Growing L^p Profiles, Random Frequency Banks, and Walsh Calibration

Consider the extremal characteristic-function dependence functional \(D_\infty(P)=\sup_t|\varphi_P(t)-\prod_j\varphi_{P_j}(t_j)|\). Unrestricted empirical maximization can saturate the universal extremal ceiling on generic finite samples, making a naive plug-in statistic misleading. We prove a stronger statistical limitation: no estimator of \(D_\infty\) is uniformly consistent over all laws on \([0,1]^2\) without regularity or support-complexity restrictions, although pointwise estimation is possible. We establish two consistent regularized alternatives. For a fixed full-support frequency distribution \(W\), an exact-integral \(L^{p_n}(W)\) estimator converges almost surely when \(p_n\to\infty\) and \(p_n=o(n)\). An independent random bank of \(K_n\) frequencies provides explicit finite-sample uncertainty brackets and almost-sure consistency when \(\log K_n=o(n)\) and \(p_n/\log K_n\to\infty\). We also derive near-maximum rates, an exact Gaussian Morse–Bott peak exponent and normal-Hessian determinant, and simultaneous Walsh confidence bounds. The guarantees distinguish deterministic Fourier observations, random frequency sampling and iid statistical estimation; they do not claim uniform consistency on the unrestricted model.