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The Fourier Cosine Method for Discrete Probability Distributions

Xiaoyu Shen, Fang Fang, Chengguang Liu

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.

The Fourier Cosine Method for Discrete Probability Distributions

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.
Paper Structure (18 sections, 13 theorems, 88 equations, 5 figures, 2 tables)

This paper contains 18 sections, 13 theorems, 88 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The COS method
  3. Notation.
  4. The COS method for continuous distributions
  5. The COS method for univariate discrete distributions
  6. Extension for bivariate discrete distributions
  7. Recovery of Probability Mass Function
  8. Superior convergence rates for some spectral filters
  9. The Lanczos filter
  10. The raised cosine filter
  11. The sharpened raised cosine filter
  12. The second order exponential filter
  13. Convergence of the implied moments
  14. Numerical examples and applications
  15. Numerical Tests of the Bounds for $\mathrm{K1}$
  16. ...and 3 more sections

Key Result

Theorem 2.1

Consider a discrete random variable $X$ with a finite number of possible values denoted by $\{\mathcal{X}_{1\leq m \leq M}\}$. Let $F_X$ represent the $\mathrm{CDF}$ of $X$. Without loss of generality, we assume $0 < \mathcal{X}_1 < \mathcal{X}_2 < \cdots < \mathcal{X}_M < \pi$. For any $x \in (0,\p where Besides, the convergence error can be precisely expressed as: where $p_m$ is the probabilit

Figures (5)

  • Figure 5: Bound of $\mathrm{K}_1$ with the second order exponential filter. $\alpha =-\ln(1/K^2)$
  • Figure 6: COS CDF of a two-point distribution with the raised cosine filter.
  • Figure 7: Comparison between the COS CDF and Monte Carlo simulated CDF
  • Figure 8: The COS-recovered conditional CDF of $N_T$
  • Figure 9: The COS-recovered conditional PMF of $N_T$

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • proof
  • Remark 1
  • Corollary 2.1.1
  • proof
  • Corollary 2.1.2
  • proof
  • Remark 2
  • ...and 21 more