The Fenton Distribution Solved

The moment-based Latent representation of correlated lognormal sums (Nagy, 2026, The Exact Latent Distribution of Correlated Lognormal Sums) relies on scaled moments \(c_k = m_k/k!\) that grow as \(e^{\sigma_{\max}^2 k^2/2}\). This super-exponential growth is the root cause of ill-conditioning in the Padé resummation step: the moment Toeplitz matrix has condition number \(\sim e^{O(N_P^2)}\), making the method impractical for \(\sigma_{\max} > 0.8\). The Smooth Latent Operator (Nagy, 2026, The Smooth Latent Operator: Parameter-Free Distributional Representations via Kernel Moment Recovery) mitigates this through regularization but cannot overcome the information-theoretic limit imposed by the growth rate.

We show that the moments are the Latent representation of the wrong generating function. The moments are Taylor coefficients of the moment generating function \(M(z) = E[e^{zS}]\), which has zero convergence radius for lognormal sums. By contrast, the Hermite-chaos expansion — the Wiener polynomial chaos of \(S\) in the underlying Gaussian variables — has coefficients that decay factorially. This is because the exponential function \(e^{Y}\) has a convergent Hermite series for \(Y \sim N(\mu, \sigma^2)\), with coefficients \(\sigma^k/k!\).

The resulting Hermite Latent \(\Lambda^H = \{c_\mathbf{k}^H\}\) lives in \(\ell^2\) (no Gaussian weight needed), the characteristic function is recovered by a convergent inner product (no Padé resummation needed), and the CDF follows by Fourier-cosine inversion (Fang and Oosterlee, 2008). The entire chain is closed-form and well-conditioned for all \(\sigma\).

We formalize this as a grade-3 Latent: the optimal choice of representation basis. The grade-3 Latent for lognormal sums is the Hermite-chaos basis, selected by matching the basis to the problem's generative structure (Gaussian \(\to\) exponential \(\to\) sum). The three-body problem's use of Fourier coefficients (instead of Taylor coefficients) is the same grade-3 choice, matched to the problem's periodic structure.

The resulting CDF formula — characteristic-function inversion composed from two contractions (Gaussian, then Fourier) — evaluates the distribution from the finite generative latent \((w, \mu, \Sigma) \in \mathbb{R}^{n(n+5)/2}\) with no free parameters. Evaluator choices (Gauss-Hermite for the Gaussian contraction, COS for the Fourier inversion) provide exponential convergence in both precision parameters. The decades-long difficulty was a convergence problem: the moment-based route uses a divergent evaluator (\(E[e^{tS}]\)) for the same inner product that the Hermite-COS route evaluates convergently (\(E[e^{itS}]\), with \(|e^{itS}|=1\)).