[Paper Review] Mod-discrete expansions

A. D. Barbour, Emmanuel Kowalski+1 more|ArXiv.org|Dec 9, 2009

Mathematical Dynamics and Fractals9 references4 citations

TL;DR

This paper introduces mod-discrete expansions for approximating the distributions of integer-valued random variables when convergence in law fails, using characteristic function ratios to derive higher-order Poisson–Charlier and translated Poisson approximations. The key contribution is explicit error bounds of order $O((\log\log n)^{-(r+1)/2})$ for the $r$-th order expansion, applicable to classical problems in number theory and combinatorics.

ABSTRACT

In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law cannot be expected. The setting is one in which the simplest approximation to the $n$'th random variable $X_n$ is by a particular member $R_n$ of a given family of distributions, whose variance increases with $n$. The basic assumption is that the ratio of the characteristic function of $X_n$ and that of R_n$ converges to a limit in a prescribed fashion. Our results cover a number of classical examples in probability theory, combinatorics and number theory.

Motivation & Objective

• To develop higher-order approximation expansions for integer-valued random variables when classical convergence in distribution fails.
• To generalize Hwang’s mod-Poisson convergence framework by deriving explicit error bounds without relying on asymptotic approximations.
• To establish conditions under which characteristic function ratios converge in a controlled way, enabling precise approximation of distributions.
• To extend the framework beyond Poisson approximations to arbitrary discrete distributions, provided their characteristic functions satisfy a basic decay condition.
• To apply the results to classical problems in number theory, such as the distribution of prime divisors $\omega(n)$ and $\Omega(n)$, yielding improved error estimates.

Proposed method

• Propose a basic estimate (Proposition 2.1) linking the closeness of signed measures to the closeness of their characteristic functions when sharing a common large-parameter factor.
• Use characteristic function ratios $\phi_{X_n}(\theta)/\phi_{R_n}(\theta) \to \psi(\theta)$, where $R_n$ is a reference distribution (e.g., Poisson), to derive expansions.
• Derive Poisson–Charlier expansions by expanding the limit characteristic function $\psi(\theta)$ in powers of $e^{i\theta} - 1$, leading to coefficients involving cumulants and arithmetic constants.
• Apply the method to translate Poisson approximations by shifting the mean to correct bias, using parameters derived from expansion coefficients.
• Establish error bounds in local, Kolmogorov, and total variation distances via the main theorem (Theorem 3.2), with explicit dependence on $n$ and $r$.
• Use the Euler product structure of zeta-related generating functions to compute expansion coefficients for number-theoretic examples like $\omega(n)$ and $\Omega(n)$.

Experimental results

Research questions

• RQ1 How can higher-order approximation expansions be systematically derived for integer-valued random variables when convergence in law does not hold?
• RQ2 What conditions on the characteristic function ratio ensure the existence and accuracy of mod-discrete expansions?
• RQ3 Can the framework be extended beyond Poisson approximations to other discrete distributions, and what is the role of the reference distribution’s characteristic function?
• RQ4 What are the precise error bounds for Poisson–Charlier and translated Poisson approximations in terms of $n$ and the expansion order $r$?
• RQ5 How can the method be applied to number-theoretic functions like $\omega(n)$ and $\Omega(n)$ to derive improved approximation bounds?

Key findings

• The paper establishes a general framework for mod-discrete expansions using characteristic function ratios, enabling higher-order approximations without relying on asymptotic settings.
• For the $r$-th order Poisson–Charlier expansion, the error in local and Kolmogorov distance is bounded by $O((\log\log n)^{-(r+1)/2})$, with explicit constants depending on $r$.
• For $\omega(n)$, the first-order Poisson–Charlier approximation has error $O(1/\log\log n)$, and a second-order approximation improves this to $O(1/(\log\log n)^{3/2})$.
• Translated Poisson approximations achieve better accuracy by adjusting the mean: for $\omega(n)$, the optimal parameters are $\lambda' \approx \log\log n - 2.0502$, $p \approx 0.3117$, $m=2$, yielding error $O((\log\log n)^{-3/2})$.
• The total variation distance is also of order $O((\log\log n)^{-(r+1)/2})$, as shown via the expansion’s tail norm and Proposition 2.4.
• The method applies to $\Omega(n)$ as well, yielding a translated Poisson approximation with $\lambda' \approx \log\log n + 0.5152$, $p \approx 0.5195$, $m=0$, and error $O((\log\log n)^{-3/2})$.