The Fenton Distribution: A Complete Analytical Characterization

We provide the first complete analytical characterization of the distribution of weighted sums of correlated lognormal random variables — the Fenton distribution \(\mathcal{F}(w, \mu, \sigma, C)\). The distribution is determined by the finite generative latent \((w, \mu, \Sigma) \in \mathbb{R}^{n(n+5)/2}\). Using characteristic function inversion with COS evaluation (Nagy, 2026), we derive closed-form or semi-analytical expressions for every standard distributional property:

1. Probability density function (PDF): a finite cosine series in log-space. 2. Cumulative distribution function (CDF): a finite sine series in log-space. 3. Moments: exact closed-form for all orders. 4. Cumulants: exact closed-form for all orders. 5. Skewness and kurtosis: exact closed-form. 6. Characteristic function: Gauss-Hermite quadrature of directly evaluated \(S(z)\). 7. Quantile function (VaR): root-finding on the COS CDF. 8. Expected Shortfall (CVaR): closed-form from COS coefficients. 9. Spectral risk measures: closed-form integral over the quantile function. 10. Mode: root-finding on the COS PDF derivative. 11. Median: special case of quantile function. 12. Entropy: numerical integration of the COS PDF. 13. Tail behavior: explicit asymptotic decay rates.

Items 1–2, 7–11 are finite trigonometric expressions — once the COS coefficients \(\{A_k\}\) are computed, evaluation is \(O(N)\) arithmetic. Items 3–6 are exact closed-form algebraic expressions requiring no numerical approximation. Items 12–13 involve one-dimensional numerical integration on the analytic PDF.

This paper serves as a distribution reference: a single document containing every formula needed to work with sums of correlated lognormals, analogous to what Johnson, Kotz, and Balakrishnan (1994) provide for the classical distribution families.