The Exact Latent Distribution of Correlated Lognormal Sums

The distribution of \(S = \sum_{i=1}^n w_i e^{Y_i}\), \(Y \sim \mathcal{N}(\mu, \Sigma)\), has no closed-form CDF. We prove that it admits a fully analytical Latent representation requiring zero numerical quadrature, zero Monte Carlo sampling, and zero eigendecomposition.

The construction exploits the fact that all moments of \(S\) have explicit closed forms: \(m_k = E[S^k]\) is a multinomial sum of Gaussian moment generating functions evaluated at integer points. The formal moment generating function \(M(z) = \sum_{k=0}^{\infty} (m_k/k!) z^k\) diverges for all \(z \neq 0\), but \(M(z)\) is well-defined and analytic for Re\((z) < 0\), with \(M(it) = \phi_S(t)\) (the characteristic function) on the imaginary axis. Since \(E[e^{-tS}]\) is completely monotone, the diagonal Padé approximant \([N_P/N_P]\) of \(M(z)\) at \(z = 0\) converges to \(M(z)\) on the left half-plane (Baker and Graves-Morris, 1996), recovering the characteristic function as a rational function — despite the divergent Taylor series.

The Padé CF is then inverted via the Fourier-cosine method (Fang and Oosterlee, 2008) to produce the Spectral Lognormal Distribution: a finite set of cosine coefficients encoding the CDF, PDF, VaR, ES, and all spectral risk measures.

The chain Moments → Padé → CF → COS → CDF is the distributional analogue of the chain Galerkin → Generating Function → Padé → Trajectory that solved the three-body problem (Nagy, 2026g). In both cases, the defining equations are algebraic, every finite truncation targets an exact analytic object, and the truncation is therefore eliminable. The sum of correlated lognormals is "solved" in the same sense as the three-body problem: an exact, finite, implicit characterization exists, with exponential convergence guarantees.