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A Fast and Accurate Numerical Method for the Left Tail of Sums of Independent Random Variables

Nadhir Ben Rached, Håkon Hoel, Johannes Vincent Meo

TL;DR

The article presents a deterministic numerical method to estimate left-tail probabilities for sums of non-negative independent random variables by iteratively convolving the density using Newton–Cotes quadrature and exploiting the periodicity of convoluted densities to apply the trapezoidal rule. It provides rigorous error and cost analyses, comparing direct convolution with FFT-based convolution, and demonstrates robustness and accuracy across diverse distributions, including Log-Normal and Lévy, as well as known and unknown sum distributions. The work shows that direct convolution often yields smaller relative errors for rare events, while FFT-based methods can be faster at larger resolutions but are more sensitive to rounding errors, and it validates the approach through extensive numerical experiments with convergence rates tied to the smoothness of the density at zero. The method offers a flexible, generic tool for left-tail rare-event estimation in wireless communications, finance, and risk, with potential extensions to multi-dimensional settings and right-tail probabilities.

Abstract

We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons-Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton-Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency-benefits to using FFT, Newton-Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method's performance and support our theoretical findings.

A Fast and Accurate Numerical Method for the Left Tail of Sums of Independent Random Variables

TL;DR

The article presents a deterministic numerical method to estimate left-tail probabilities for sums of non-negative independent random variables by iteratively convolving the density using Newton–Cotes quadrature and exploiting the periodicity of convoluted densities to apply the trapezoidal rule. It provides rigorous error and cost analyses, comparing direct convolution with FFT-based convolution, and demonstrates robustness and accuracy across diverse distributions, including Log-Normal and Lévy, as well as known and unknown sum distributions. The work shows that direct convolution often yields smaller relative errors for rare events, while FFT-based methods can be faster at larger resolutions but are more sensitive to rounding errors, and it validates the approach through extensive numerical experiments with convergence rates tied to the smoothness of the density at zero. The method offers a flexible, generic tool for left-tail rare-event estimation in wireless communications, finance, and risk, with potential extensions to multi-dimensional settings and right-tail probabilities.

Abstract

We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons-Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton-Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency-benefits to using FFT, Newton-Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method's performance and support our theoretical findings.
Paper Structure (16 sections, 8 theorems, 82 equations, 5 figures, 6 tables)

This paper contains 16 sections, 8 theorems, 82 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The numerical method
  3. Numerical integration
  4. Efficient computation of bar f conv
  5. Implementations of discrete linear convolution
  6. Theoretical results
  7. Rounding errors
  8. Computational cost
  9. Numerical experiments
  10. Comparison of direct- and FFT-based convolution
  11. Log-Normal distribution
  12. Lévy distribution
  13. Estimating known distributions
  14. Convergence properties for the convolution method
  15. Estimating unknown distributions
  16. ...and 1 more sections

Key Result

Lemma 1

Let $f \in C_{0}^{2p}([0,\gamma])$ for some integer $p \ge 1$ and let $f^{\circledast n}(x_k)$ for $n\ge 2$ be defined by eq:associativeDiscreteConvProp. Then, there exists a constant $C_1>0$ that depends on $p$ such that where with the conventions that $\prod_{\ell =2}^{m-1} f(x_{j_\ell}) \equiv 1$ when $m =2$ and $f^{*1} = f$.

Figures (5)

  • Figure 1: Left: The probability density function $p(y) = \bar{f}^{\circledast 16}(y)$ for direct convolution and FFT-based convolution for the rare-event problem studied in Section \ref{['sec:convVsFFT']}. Right: The runtime for direct convolution and FFT-based convolution for the rare-event problem studied in Section \ref{['sec:convVsFFT']}.
  • Figure 2: The probability density function to $Y = \sum_{i=1}^{16}X_i$ where $X_i$ are i.i.d. Lévy distributed RVs. The density is computed by the exact formula, by direct convolution and FFT-based convolution.
  • Figure 3: Left: Relative error as a function of the mesh-size when estimating $F_Y(0.8), Y \sim \chi^2(16df)$. Right: Relative error as a function of the mesh-size when we are estimating $F_Y(0.8), Y \sim \text{Lévy} \left (0, \left(\sum_{i=1}^n \sqrt{c_i} \right)^2 \right )$
  • Figure 4: Left: Relative error as a function of the mesh-size when approximating $\alpha_{N_M}$ in the case when $X_i \sim \text{Log-Normal}(0, 0.125)$ where ($\alpha_{N_M}$ is a pseudo-reference solution calculated using the convolution method with $N_M = 1e6$ using Boole's rule in the last step. Right: Relative error as a function of the mesh-size when estimating pseudo-reference solution $\alpha_{N_M}$ of the CDF of a sum of Log-Normals with varying $\sigma$ calculated using the convolution method with $N_M = 1e6$ using Boole's rule in the last step
  • Figure 5: Left: Relative error as a function of the mesh-size when estimating $F_Y(0.8)$ where $Y = \sum_{i=1}^{16} X_i$ and $X_i \sim \text{Nakagami-m}(m)$. Right: Relative error as a function of the mesh-size when estimating $F_Y(0.8)$ where $Y = \sum_{i=1}^{16} X_i$ and $X_i \sim \text{Rice}(\nu)$

Theorems & Definitions (19)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 4
  • Lemma 2: Rounding error direct convolution
  • proof
  • ...and 9 more