Bernoulli

  • Bernoulli
  • Volume 19, Number 3 (2013), 982-1005.

Properties and numerical evaluation of the Rosenblatt distribution

Mark S. Veillette and Murad S. Taqqu

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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 Lévy–Khintchine formula and derive asymptotic properties of the Lévy 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.

Article information

Source
Bernoulli, Volume 19, Number 3 (2013), 982-1005.

Dates
First available in Project Euclid: 26 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1372251150

Digital Object Identifier
doi:10.3150/12-BEJ421

Mathematical Reviews number (MathSciNet)
MR3079303

Zentralblatt MATH identifier
1273.60020

Keywords
Edgeworth expansions long range dependence Rosenblatt distribution self-similarity

Citation

Veillette, Mark S.; Taqqu, Murad S. Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli 19 (2013), no. 3, 982--1005. doi:10.3150/12-BEJ421. https://projecteuclid.org/euclid.bj/1372251150


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Supplemental materials

  • Supplementary material: Supplement to Properties and numerical evaluation of the Rosenblatt distribution. The supplement [35] to this article details the approximation of the integral operator $\mathcal{K}_{D}$ and the computation of the cumulants, moments and CDF of $Z_{D}$. It also contains an extensive table of the CDF of $Z_{D}$ and a guide to the software.