Properties and numerical evaluation of the Rosenblatt distribution

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.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ421 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Limit behavior of the Rosenblatt Ornstein–Uhlenbeck process with respect to the Hurst index

“…• The above result shows that, when H → 1 2 , the Wiener-Rosenblatt integral R f (u)dZ H (u) converges in distribution to R f (u)dW (u), where W is a Wiener process. This is a natural extension of the results in [1] or [15].…”

“…and implies (15) and consequently, it gives the convergence of finite dimensional distributions of Y H as → 1 2 . The tighness follows (24).…”

“…There are several recent research works that investigate the asymptotic behavior in distribution of some fractional processes (see [3], [2], [1], [15]) with respect to the Hurst parameter. In particular, in the case of the Rosenblatt process (Z H (t)) t≥0 with self-similarity index H ∈ ( 1 2 , 1), it has been shown in [15] that Z H converges weakly, as H → 1 2 , in the space of continuous functions C[0, T ] (for every T > 0), to a Brownian motion while if H → 1, it tends weakly to the stochastic process (t 1 √ 2 (Z 2 − 1)) t≥0 , Z 2 − 1 being a so-called centered chi-square random variable. The case of the generalized Rosenblatt process has been considered in [2] while the case of the Rosenblatt sheet can be found in [1].…”

From intersection local time to the Rosenblatt process

“…Infinite divisibility was recently proved in [23,40]. There does not seem to be information in the literature on whether or not the Rosenblatt process has the Markov property (it seems plausible that it does not, by analogy with fractional Brownian motion).…”

“…Significant attention has been attracted by the Rosenblatt process due to its mathematical interest, and possible applications where the Gaussian property may not be assumed. Recent papers on the subject include [39], which develops a related stochastic calculus and mentions areas of application (see also references therein), [23] and [40], where new properties of the process have been found, and [17], which provides a strong approximation for the process. Relevant information for the present paper on the Rosenblatt process and fractional Brownian motion is given in the next section.…”