A tail bound for cumulant series for complex functions of independent random variables

TL;DR

This paper derives explicit bounds on the truncation error of cumulant series for bounded complex functions of independent random variables, extending previous real-valued results and enabling applications in probability approximations and combinatorics.

Contribution

It extends cumulant series bounds from real to complex functions of independent variables using multidimensional differences, broadening theoretical and practical applications.

Findings

Provides explicit truncation error bounds for complex functions

Extends previous real-valued cumulant bounds to complex functions

Demonstrates applications in Berry--Esseen bounds, graph enumeration, and Edgeworth expansions

Abstract

We obtain explicit bounds on the truncation error of the cumulant series of a bounded complex function of a random vector with independent components. The bounds are based on multidimensional differences. This extends the theory of the author with Brendan McKay and Rui-Ray Zhang (J. Combin. Th., Ser. B, 2025) from real functions to complex functions. We demonstrate some initial applications including a Berry--Esseen bound, an Edgeworth expansion for triangles in random graphs, and enumeration of regular graphs.