Abstract:
A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in d-dimensional space. In the particular case of a non-linear function of a chi-squared random field with Laguerre rank equal to one, we apply the Karhunen-Lo\'eve expansion and the Fredholm determinant formula to obtain the characteristic function of its Rosenblatt-type limit distribution. When the Laguerre rank equals one and two, we obtain the multiple Wiener-It\^o stochastic integral representation of the limit distribution. In both cases, an infinite series representation in terms of independent random variables is constructed for the limit random variables.

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

Abstract:
Long-range dependence in time series may yield non-central limit theorems. We show that there are analogous time series in free probability with limits represented by multiple Wigner integrals, where Hermite processes are replaced by non-commutative Tchebycheff processes. This includes the non-commutative fractional Brownian motion and the non-commutative Rosenblatt process.

Abstract:
We start with an i.i.d.\ sequence and consider the product of two polynomial-forms moving averages based on that sequence. The coefficients of the polynomial forms are asymptotically slowly decaying homogeneous functions so that these processes have long memory. The product of these two polynomial forms is a stationary nonlinear process. We want to obtain limit theorems for the normalized sums of this nonlinear process in two cases: exclusion of the diagonal terms of the polynomial form, or inclusion. In either case, if the product has long memory, then the limits are given by a Wiener chaos, but these limits are quite different. If the diagonals are excluded, then the limit is expressed as in the product formula of two Wiener-It\^o integrals. When the diagonals are included, the limit stochastic integrals are typically due to a single factor of the product, namely the one with the strongest memory.

Abstract:
Although fractional Brownian motion was not invented by Benoit Mandelbrot, it was he who recognized the importance of this random process and gave it the name by which it is known today. This is a personal account of the history behind fractional Brownian motion and some subsequent developments.

Abstract:
Novel thin-disk laser pump layouts are proposed yielding an increased number of passes for a given pump module size and pump source quality. These novel layouts result from a general scheme which bases on merging two simpler pump optics arrangements. Some peculiar examples can be realized by adapting standard commercially available pump optics simply by intro ducing an additional mirror-pair. More pump passes yield better efficiency, opening the way for usage of active materials with low absorption. In a standard multi-pass pump design, scaling of the number of beam passes brings ab out an increase of the overall size of the optical arrangement or an increase of the pump source quality requirements. Such increases are minimized in our scheme, making them eligible for industrial applications

Abstract:
Consider generalized adapted stochastic integrals with respect to independently scattered random measures with second moments. We use a decoupling technique, known as the "principle of conditioning", to study their stable convergence towards mixtures of infinitely divisible distributions. Our results apply, in particular, to multiple integrals with respect to independently scattered and square integrable random measures, as well as to Skorohod integrals on abstract Wiener spaces. As a specific application, we establish a Central Limit Theorem for sequences of double integrals with respect to a general Poisson measure, thus extending the results contained in Nualart and Peccati (2005) and Peccati and Tudor (2004) to a non-Gaussian context.

Abstract:
We prove sufficient conditions, ensuring that a sequence of multiple Wiener-It\^{o} integrals (with respect to a general Gaussian process) converges stably to a mixture of normal distributions. Our key tool is an asymptotic decomposition of contraction kernels, realized by means of increasing families of projection operators. We also use an infinite-dimensional Clark-Ocone formula, as well as a version of the correspondence between "abstract" and "concrete" filtered Wiener spaces, in a spirit similar to \"{U}st\"{u}nel and Zakai (1997).

Abstract:
We study the limit law of a vector made up of normalized sums of functions of long-range dependent stationary Gaussian series. Depending on the memory parameter of the Gaussian series and on the Hermite ranks of the functions, the resulting limit law may be (a) a multivariate Gaussian process involving dependent Brownian motion marginals, or (b) a multivariate process involving dependent Hermite processes as marginals, or (c) a combination. We treat cases (a), (b) in general and case (c) when the Hermite components involve ranks 1 and 2. We include a conjecture about case (c) when the Hermite ranks are arbitrary.