We consider the solution $\{u(t, x); t≥0,x∈\mathbf R\}$ of a system of $d$ linear stochastic wave equations driven by a $d$-dimensional symmetric space-time Lévy noise. We provide a necessary and sufficient condition on the characteristic exponent of the Lévy noise, which describes exactly when the zero set of $u$ is non-void. We also compute the Hausdorff dimension of that zero set when it is non-empty. These results will follow from more general potential-theoretic theorems on the level sets of Lévy sheets.

We obtain dimension-free concentration inequalities for $ℓ^p$-norms, $p≥2$, of infinitely divisible random vectors with independent coordinates and finite exponential moments. Besides such norms, the methods and results extend to some other classes of Lipschitz functions.

The simple Galton–Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of the family tree. This viewpoint has led to new insights and a revival of classical theory.

We show how a similar reinterpretation can shed new light on the more interesting forms of branching processes that allow repeated bearings and, thus, overlapping generations. In particular, we use the stable pedigree law to give a transparent description of a size-biased version of general branching processes in discrete time. This allows us to analyze the $x \log x$ condition for exponential growth of supercritical general processes as well as relation between simple Galton–Watson and more general branching processes.

Invariance principles are obtained for a Markov process on a half-line with continuous paths on the interior. The domains of attraction of the two different types of self-similar processes are investigated. Our approach is to establish convergence of excursion point processes, which is based on Itô’s excursion theory and a recent result on convergence of excursion measures by Fitzsimmons and the present author.

We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous’ beta-splitting model, which has an extended parameter range $β>−2$ with respect to the beta $(β+1, β+1)$ probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter Poisson–Dirichlet models for exchangeable random partitions of $ℕ$, with an extended parameter range $0≤α≤1$, $θ≥−2α$ and $α<0$, $θ=−mα$, $m∈ℕ$.

In the world of multivariate extremes, estimation of the dependence structure still presents a challenge and an interesting problem. A procedure for the bivariate case is presented that opens the road to a similar way of handling the problem in a truly multivariate setting. We consider a semi-parametric model in which the stable tail dependence function is parametrically modeled. Given a random sample from a bivariate distribution function, the problem is to estimate the unknown parameter. A method of moments estimator is proposed where a certain integral of a nonparametric, rank-based estimator of the stable tail dependence function is matched with the corresponding parametric version. Under very weak conditions, the estimator is shown to be consistent and asymptotically normal. Moreover, a comparison between the parametric and nonparametric estimators leads to a goodness-of-fit test for the semiparametric model. The performance of the estimator is illustrated for a discrete spectral measure that arises in a factor-type model and for which likelihood-based methods break down. A second example is that of a family of stable tail dependence functions of certain meta-elliptical distributions.

The multivariate extremal index function relates the asymptotic distribution of the vector of pointwise maxima of a multivariate stationary sequence to that of the independent sequence from the same stationary distribution. It also measures the degree of clustering of extremes in the multivariate process. In this paper, we construct nonparametric estimators of this function and prove their asymptotic normality under long-range dependence and moment conditions. The results are illustrated by means of a simulation study.

Let $(X, Y)$ be a random vector whose conditional excess probability $θ(x, y):=P(Y≤y | X>x)$ is of interest. Estimating this kind of probability is a delicate problem as soon as $x$ tends to be large, since the conditioning event becomes an extreme set. Assume that $(X, Y)$ is elliptically distributed, with a rapidly varying radial component. In this paper, three statistical procedures are proposed to estimate $θ(x, y)$ for fixed $x, y$, with $x$ large. They respectively make use of an approximation result of Abdous et al. (cf. Canad. J. Statist. 33 (2005) 317–334, Theorem 1), a new second order refinement of Abdous et al.’s Theorem 1, and a non-approximating method. The estimation of the conditional quantile function $θ(x, ⋅)^←$ for large fixed $x$ is also addressed and these methods are compared via simulations. An illustration in the financial context is also given.

We propose a procedure to handle the problem of Gaussian regression when the variance is unknown. We mix least-squares estimators from various models according to a procedure inspired by that of Leung and Barron [IEEE Trans. Inform. Theory 52 (2006) 3396–3410]. We show that in some cases, the resulting estimator is a simple shrinkage estimator. We then apply this procedure to perform adaptive estimation in Besov spaces. Our results provide non-asymptotic risk bounds for the Euclidean risk of the estimator.

Stute [Ann. Probab. 19 (1991) 812–825] introduced a class of estimators called conditional $U$-statistics. They can be seen as a generalization of the Nadaraya–Watson estimator for the regression function. Stute proved their strong pointwise consistency to $$m(\mathbf{t}):=\mathbb{E}[g(Y_{1},\ldots,Y_{m})|(X_{1},\ldots,X_{m})=\mathbf{t}],\qquad\mathbf{t}\in\mathbb{R}^{m}.$$ Very recently, Giné and Mason introduced the notion of a local $U$-process, which generalizes that of a local empirical process, and obtained central limit theorems and laws of the iterated logarithm for this class. We apply the methods developed in Einmahl and Mason [Ann. Statist. 33 (2005) 1380–1403] and Giné and Mason [Ann. Statist. 35 (2007) 1105–1145; J. Theor. Probab. 20 (2007) 457–485] to establish uniform in t and in bandwidth consistency to m(t) of the estimator proposed by Stute. We also discuss how our results are used in the analysis of estimators with data-dependent bandwidths.

A function $ρ:[0, ∞)→(0, 1]$ is a completely monotonic function if and only if $ρ(‖\mathbf{x}‖^2)$ is positive definite on $ℝ^d$ for all $d$ and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function $$\rho(\beta ,\gamma)(x)=1-\biggl(\frac{x^{\beta}}{1+x^{\beta}}\biggr)^{\gamma},\qquad x\ge 0,\ \beta,\gamma>0,$$ called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any $d$-dimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.

In this paper, we study the problem of pointwise estimation of a multivariate function. We develop a general pointwise estimation procedure that is based on selection of estimators from a large parameterized collection. An upper bound on the pointwise risk is established and it is shown that the proposed selection procedure specialized for different collections of estimators leads to minimax and adaptive minimax estimators in various settings.