The Fourier Cosine Method for Discrete Probability Distributions

ArXiv ID: 2410.04487 “View on arXiv”

Authors: Unknown

Abstract

We provide a rigorous convergence proof demonstrating that the well-known semi-analytical Fourier cosine (COS) formula for the inverse Fourier transform of continuous probability distributions can be extended to discrete probability distributions, with the help of spectral filters. We establish general convergence rates for these filters and further show that several classical spectral filters achieve convergence rates one order faster than previously recognized in the literature on the Gibbs phenomenon. Our numerical experiments corroborate the theoretical convergence results. Additionally, we illustrate the computational speed and accuracy of the discrete COS method with applications in computational statistics and quantitative finance. The theoretical and numerical results highlight the method’s potential for solving problems involving discrete distributions, particularly when the characteristic function is known, allowing the discrete Fourier transform (DFT) to be bypassed.

Keywords: COS Method (Fourier Cosine), Discrete Probability Distributions, Spectral Filters, Gibbs Phenomenon, Characteristic Functions, Quantitative Finance (General)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper features advanced mathematical proofs involving spectral filters, Gibbs phenomenon, and convergence rates, indicating high math complexity. While it includes numerical experiments and applications in quantitative finance, the emphasis is on theoretical convergence proofs rather than extensive backtesting or real-world data implementation, resulting in moderate empirical rigor.

  flowchart TD
    A["Research Goal: Extend COS Method to Discrete Distributions"] --> B["Methodology: Extend with Spectral Filters"]
    B --> C["Input: Characteristic Function of Discrete Distribution"]
    C --> D["Computational Process: Discrete COS Method"]
    D --> E["Outcome 1: Rigorous Convergence Proof & Fast Rates"]
    D --> F["Outcome 2: Bypassing DFT for Speed"]
    E --> G["Validation: Numerical Experiments & Applications"]
    F --> G