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The Karhunen-Lo`eve expansion and the Fredholm determinant formula are used, to derive\ud an asymptotic Rosenblatt-type distribution of a sequence of integrals of quadratic functions of\ud Gaussian stationary random fields on R\ud d displaying long-range dependence. This distribution\ud reduces to the usual Rosenblatt distribution when d = 1. Several properties of this new distribution\ud are obtained. Specifically, its series representation, in terms of independent chi-squared\ud random variables, is established. Its L´evy-Khintchine representation, and membership to the\ud Thorin subclass of self-decomposable distributions are obtained as well. The existence and\ud boundedness of its probability density then follow as a direct consequence.

It has become common practice to use heavy-tailed distributions in order to describe the variations in time and space of network traffic workloads. The asymptotic behavior of these workloads is complex; different limit processes emerge depending on the specifies of the work arrival structure and the nature of the asymptotic scaling. We focus on two variants of the infinite source Poisson model and provide a coherent and unified presentation of the scaling theory by using integral representations. This allows us to understand physically why the various limit processes arise.

We consider time series $Y_t = G(X_t)$ where $X_t$ is Gaussian with long memory and $G$ is a polynomial. The series $Y_t$ may or may not have long memory. The spectral density $g_\theta(x)$ of $Y_t$ is parameterized by a vector $\theta$ and we want to estimate its true value $\theta_0$ . We use a least-squares Whittle-type estimator $\hat{\theta}_N$ for $\theta_0$, based on observations $Y_1,\dots,Y_N$. If $Y_t$ is Gaussian, then $\sqrt{N}(\hat{\theta}_N-\theta_0)$ converges to a Gaussian distribution. We show that for non-Gaussian time series $Y_t$ , this $\sqrt{N}$ consistency of the Whittle estimator does not always hold and that the limit is not necessarily Gaussian. This can happen even if $Y_t$ has short memory.