Properties and numerical evaluation of the Rosenblatt distribution

TL;DR

This paper thoroughly analyzes the Rosenblatt distribution, providing methods for computing its properties, including cumulants, moments, and density functions, with high precision and supporting software tools.

Contribution

It introduces a novel technique for cumulant computation and expands the distribution in terms of chi-squared distributions, enhancing understanding and calculation accuracy.

Findings

Derived the cumulants, moments, and density functions with high precision

Developed a chi-squared expansion for the Rosenblatt distribution

Provided software tools for practical implementation

Abstract

This paper studies various distributional properties of the Rosenblatt distribution. We begin by describing a technique for computing the cumulants. We then study the expansion of the Rosenblatt distribution in terms of shifted chi-squared distributions. We derive the coefficients of this expansion and use these to obtain the L\'{e}vy-Khintchine formula and derive asymptotic properties of the L\'{e}vy measure. This allows us to compute the cumulants, moments, coefficients in the chi-square expansion and the density and cumulative distribution functions of the Rosenblatt distribution with a high degree of precision. Tables are provided and software written to implement the methods described here is freely available by request from the authors.