When Simulation Is Unnecessary: An Information-Theoretic Characterization of Analytically Tractable Distributions

Monte Carlo simulation exists because we cannot compute integrals analytically. But for how large a class of distributions can we compute them? We characterize this class precisely via a single number: the analyticity radius \(\rho\). For any distribution with \(\rho > 1\), \(N = \lceil\log(C_f/(\varepsilon(1-\rho^{-1})))/\log\rho\rceil\) Fourier coefficients suffice to compute every expectation, quantile, and coherent risk measure to accuracy \(\varepsilon\) — deterministically, without simulation. For a typical lognormal portfolio with \(\rho \approx 1.1\) and accuracy \(\varepsilon = 10^{-6}\), this gives \(N \approx 145\) (see Section 2.3 for the derivation). The number \(N\) depends on accuracy and smoothness, not on the complexity of the underlying model or the number of random variables. We show that: (i) convolution preserves analyticity and \(\rho \ge \min(\rho_1, \rho_2)\) (diversification = smoothing), (ii) diffusion guarantees \(\rho > 1\) (any SDE with \(\sigma > 0\) produces spectrally tractable marginals), (iii) Bayesian updating in the spectral domain is an \(O(N^2)\) convolution (no MCMC), (iv) the boundary \(\rho = 1\) separates tractable from intractable — distributions with jumps, atoms, or non-analytic densities require simulation. The result provides a decision criterion for practitioners: compute \(\rho\) for your distribution; if \(\rho > 1\), Monte Carlo is unnecessary. For financial portfolios under standard models: \(\rho \in [1.1, 3.0]\), giving \(N \in [40, 200]\). The complexity comparison is \(O(\log(1/\varepsilon)/\log\rho)\) versus \(O(1/\varepsilon^2)\): for \(\varepsilon = 10^{-6}\) and \(\rho = 1.1\), the spectral method requires \(\sim 145\) terms versus \(\sim 10^{12}\) Monte Carlo samples for comparable accuracy, a gap of many orders of magnitude (see Section 11.3 for explicit crossover tables). The key results — including the convergence barrier theorem and branching irrelevance principle — are formally verified in Lean 4.