Asymptotic universal moment matching properties of normal distributions

ArXiv ID: 2508.03790 “View on arXiv”

Authors: Xuan Liu

Abstract

Moment matching is an easy-to-implement and usually effective method to reduce variance of Monte Carlo simulation estimates. On the other hand, there is no guarantee that moment matching will always reduce simulation variance for general integration problems at least asymptotically, i.e. when the number of samples is large. We study the characterization of conditions on a given underlying distribution $X$ under which asymptotic variance reduction is guaranteed for a general integration problem $\mathbb{“E”}[“f(X)”]$ when moment matching techniques are applied. We show that a sufficient and necessary condition for such asymptotic variance reduction property is $X$ being a normal distribution. Moreover, when $X$ is a normal distribution, formulae for efficient estimation of simulation variance for (first and second order) moment matching Monte Carlo are obtained. These formulae allow estimations of simulation variance as by-products of the simulation process, in a way similar to variance estimations for plain Monte Carlo. Moreover, we propose non-linear moment matching schemes for any given continuous distribution such that asymptotic variance reduction is guaranteed.

Keywords: Monte Carlo Simulation, Variance Reduction, Moment Matching, Asymptotic Variance, Integration Problems, Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper presents highly advanced, theoretical mathematics (e.g., asymptotic analysis, characterization proofs) for variance reduction in Monte Carlo simulation but lacks any backtesting, real data, or implementation details, focusing purely on theoretical properties.

  flowchart TD
    A["Research Goal:<br>Find conditions for guaranteed<br>asymptotic variance reduction<br>using Moment Matching (MM)"] --> B["Methodology:<br>Mathematical derivation & analysis<br>of MM for general integration<br>𝔼[f(X)"]]
    B --> C["Input: Underlying Distribution X"]
    C --> D["Computational Process:<br>Analyze Asymptotic Variance<br>of MM estimator for X"]
    D --> E{"Condition Check:<br>Does X reduce variance?"}
    E -- Necessary & Sufficient Condition --> F["Key Finding 1:<br>X must be a Normal Distribution"]
    E -- Special Case --> G["Key Finding 2:<br>Formulas for efficient variance<br>estimation in Normal case"]
    F --> H["Proposed Outcome:<br>Non-linear MM schemes for<br>any continuous distribution<br>to guarantee variance reduction"]