In this talk we discuss the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has in nite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. In the lighter-tailed case, we study the asymptotics of the scaled position of the rightmost particle in the n-th generation and show the existence of a non-trivial constant. This is a joint work with Souvik Ray (Stanford), Parthanil Roy (ISI, Bangalore) and Philippe Soulier (Universite Paris Nanterre).[-]

In this talk, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving Lévy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the Lévy process is weaker. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander-Rosenblatt statistic and the Cramér-von Mises statistic.[-]

The Kingman coalescent is a fundamental process in population genetics modelling the ancestry of a sample of individuals backwards in time. In this paper, weak convergence is proved for a sequence of Markov chains consisting of two components related to the Kingman coalescent, under a parent dependent d-alleles mutation scheme, as the sample size, grows to infinity. The first component is the normalised d-dimensional jump chain of the block counting processes of the Kingman coalescent. The second component is a d^2-dimensional process counting the number of mutations between types occurring in the Kingman coalescent. Time is scaled by the sample size. The limiting process consists of a deterministic d-dimensional component, describing the limit of the block counting jump chain, and d^2 independent Poisson processes with state-dependent intensities, exploding at the origin, describing the limit of the number of mutations. The weak convergence result is first proved, using a generator approach, in the setting of parent independent mutations. A change of measure argument is used to extend the weak convergence result to include parent dependent mutations.[-]

We study random fields taking values in a separable Hilbert space H. First, we focus on their local structure and establish a counterpart to Falconer's characterization of tangent fields. That is, we show (under general conditions) that the tangent fields to a H-valued process are self-similar and almost all of them have stationary increments. We go a bit further and study higher-order tangent fields. This leads naturally to the study of self-similar intrinsic random functions (IRF) taking values in a Hilbert space. To this end, we begin by extending Matheron's theory of scalar-valued IRFs and provide the spectral representation of H-valued IRFs. We then use this theory to characterize large classes of operator self-similar H-valued IRF processes, which in the Gaussian case can be viewed as the H-valued counterparts to fractional Brownian fields. These general results may find applications to the study of long-range dependence for random fields taking values in a Hilbert space as well as to modeling function-valued spatial data.[-]

We consider the monogenic representation for self-similar random fields. This approach is based on the monogenic representation of a greyscale image, using Riesz transform, and is particularly well-adapted to detect directionality of self-similar Gaussian fields. In particular, we focus on distributions of monogenic parameters defined as amplitude, orientation and phase of the spherical coordinates of the wavelet monogenic representation. This allows us to define estimators for some anisotropic fractional fields. We then consider the elliptical monogenic model to define vector-valued random fields according to natural colors, using the RGB color model. Joint work with Philippe Carre (XLIM, Poitiers), Céline Lacaux (LMA, Avignon) and Claire Launay (IDP, Tours).[-]