Properties and numerical evaluation of the Rosenblatt distribution

By Mark S. Veillette, Murad S. Taqqu. 2013

98 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/00f59b328dcc3673fbfc3229296b063ce3fd32b1

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

In Statistics and Probability

Citations in the corpus

Generalized Hermite processes, discrete chaos and limit theorems (2013, methodology): “(Proposition 9.3.1 of Peccati and Taqqu [23]) Let gj(x) be a real-valued function in L2(Rkj ), j = 1, . . . , J .”
On Stein's Method for Infinitely Divisible Laws with Finite First Moment (2017, background): “∞. In particular, the Rosenblatt distribution belongs to this class of second Wiener chaos random variables, since asymptotically λk ∼ k→+∞ CDkD−1, for some CD>0 and D∈ (0,1/2) (see Theorem 3.2 of [39]). As seen in some of the examples presented above, characterizing identities can involve local ”
Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter (2017, background): “Readers who are interested in the density plot of Rosenblatt random variable are referred to the paper (Veillette and Taqqu 2013) and the references therein.”