We consider a symmetric, finite-range contact process with two types of infection; both have the same (supercritical) infection rate and heal at rate 1, but sites infected by Infection 1 are immune to Infection 2. We take the initial configuration where sites in $(−∞, 0]$ have Infection 1 and sites in $[1, ∞)$ have Infection 2, then consider the process $ρ_t$ defined as the size of the interface area between the two infections at time $t$. We show that the distribution of $ρ_t$ is tight, thus proving a conjecture posed by Cox and Durrett in [Bernoulli 1 (1995) 343–370].

Let $X = \{X(t), t ∈ ℝ^N\}$ be a random field with values in $ℝ^d$. For any finite Borel measure $μ$ and analytic set $E ⊂ ℝ^N$, the Hausdorff and packing dimensions of the image measure $μ_X$ and image set $X(E)$ are determined under certain mild conditions. These results are applicable to Gaussian random fields, self-similar stable random fields with stationary increments, real harmonizable fractional Lévy fields and the Rosenblatt process.

Ball throwing on Euclidean spaces has been considered for some time. A suitable renormalization leads to a fractional Brownian motion as limit object. In this paper, we investigate ball throwing on spheres. A different behavior is exhibited: we still get a Gaussian limit, but it is no longer a fractional Brownian motion. However, the limit is locally self-similar when the self-similarity index $H$ is less than $1/2$.

We study conditions under which $$\mathbf{P}\{S_τ >x\} ∼ \mathbf{P}\{M_τ > x\} ∼ \mathbf{E}τ\mathbf{P}\{ξ_1 > x\} \mbox{as } x → ∞,$$ where $S_τ$ is a sum $ξ_1 + ⋯ + ξ_τ$ of random size $τ$ and $M_τ$ is a maximum of partial sums $M_τ = \max_{n≤τ} S_n$. Here, $ξ_n, n = 1, 2, …,$ are independent identically distributed random variables whose common distribution is assumed to be subexponential. We mostly consider the case where $τ$ is independent of the summands; also, in a particular situation, we deal with a stopping time.

We also consider the case where $\mathbf{E}ξ > 0$ and where the tail of $τ$ is comparable with, or heavier than, that of $ξ$, and obtain the asymptotics $$\mathbf{P}\{S_τ > x\} ∼ \mathbf{E}τ\mathbf{P}\{ξ_1 > x\} + \mathbf{P}\{τ > x / \mathbf{E}ξ\} \mbox{as } x → ∞.$$ This case is of primary interest in branching processes.

In addition, we obtain new uniform (in all $x$ and $n$) upper bounds for the ratio $\mathbf{P}\{S_n > x\} / \mathbf{P}\{ξ_1 > x\}$ which substantially improve Kesten’s bound in the subclass ${\mathcal{S}}^{*}$ of subexponential distributions.

In this paper, we study the weak convergence of the integrated periodogram indexed by classes of functions for linear processes with symmetric $α$-stable innovations. Under suitable summability conditions on the series of the Fourier coefficients of the index functions, we show that the weak limits constitute $α$-stable processes which have representations as infinite Fourier series with i.i.d. $α$-stable coefficients. The cases $α ∈ (0, 1)$ and $α ∈ [1, 2)$ are dealt with by rather different methods and under different assumptions on the classes of functions. For example, in contrast to the case $α ∈ (0, 1)$, entropy conditions are needed for $α ∈ [1, 2)$ to ensure the tightness of the sequence of integrated periodograms indexed by functions. The results of this paper are of additional interest since they provide limit results for infinite mean random quadratic forms with particular Toeplitz coefficient matrices.

In this paper, we deal with the asymptotic distribution of the maximum increment of a random walk with a regularly varying jump size distribution. This problem is motivated by a long-standing problem on change point detection for epidemic alternatives. It turns out that the limit distribution of the maximum increment of the random walk is one of the classical extreme value distributions, the Fréchet distribution. We prove the results in the general framework of point processes and for jump sizes taking values in a separable Banach space.

In this paper, we investigate the problem of nonparametric monotone frontier estimation from the perspective of extreme value theory. This enables us to revisit the asymptotic theory of the popular free disposal hull estimator in a more general setting, to derive new and asymptotically Gaussian estimators and to provide useful asymptotic confidence bands for the monotone boundary function. The finite-sample behavior of the suggested estimators is explored via Monte Carlo experiments. We also apply our approach to a real data set based on the production activity of the French postal services.

Let $(X_t)_{t∈T}$ be a family of real-valued centered random variables indexed by a countable set $T$. In the first part of this paper, we establish exponential bounds for the deviation probabilities of the supremum $Z = \sup_{t∈T} X_t$ by using the generic chaining device introduced in Talagrand (Inst. Hautes Études Sci. Publ. Math. 81 (1995) 73–205). Compared to concentration-type inequalities, these bounds offer the advantage of holding under weaker conditions on the family $(X_t)_{t∈T}$. The second part of the paper is oriented toward statistics. We consider the regression setting $Y = f + ξ$, where $f$ is an unknown vector in $ℝ^n$ and $ξ$ is a random vector, the components of which are independent, centered and admit finite Laplace transforms in a neighborhood of $0$. Our aim is to estimate $f$ from the observation of $Y$ by means of a model selection approach among a collection of linear subspaces of $ℝ^n$. The selection procedure we propose is based on the minimization of a penalized criterion, the penalty of which is calibrated by using the deviation bounds established in the first part of this paper. More precisely, we study suprema of random variables of the form $X_t = ∑_{i=1}^n t_iξ_i$, where $t$ varies in the unit ball of a linear subspace of $ℝ^n$. Finally, we show that our estimator satisfies an oracle-type inequality under suitable assumptions on the metric structures of the linear spaces of the collection.

Using Bernstein polynomial approximations, we prove the central limit theorem for linear spectral statistics of sample covariance matrices, indexed by a set of functions with continuous fourth order derivatives on an open interval including $[(1-\sqrt{y})^{2},(1+\sqrt{y})^{2}]$, the support of the Marčenko–Pastur law. We also derive the explicit expressions for asymptotic mean and covariance functions.

Compound Poisson distributions and signed compound Poisson measures are used for approximation of the Markov binomial distribution. The upper and lower bound estimates are obtained for the total variation, local and Wasserstein norms. In a special case, asymptotically sharp constants are calculated. For the upper bounds, the smoothing properties of compound Poisson distributions are applied. For the lower bound estimates, the characteristic function method is used.

Given an i.i.d. sample from a distribution $F$ on $ℝ$ with uniformly continuous density $p_0$, purely data-driven estimators are constructed that efficiently estimate $F$ in sup-norm loss and simultaneously estimate $p_0$ at the best possible rate of convergence over Hölder balls, also in sup-norm loss. The estimators are obtained by applying a model selection procedure close to Lepski’s method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or $B$-splines. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-type analogs of the inequalities in Koltchinskii [Ann. Statist. 34 (2006) 2593–2656] for the deviation of suprema of empirical processes from their Rademacher symmetrizations.

We consider design issues for toxicology studies when we have a continuous response and the true mean response is only known to be a member of a class of nested models. This class of non-linear models was proposed by toxicologists who were concerned only with estimation problems. We develop robust and efficient designs for model discrimination and for estimating parameters in the selected model at the same time. In particular, we propose designs that maximize the minimum of $D$- or $D_1$-efficiencies over all models in the given class. We show that our optimal designs are efficient for determining an appropriate model from the postulated class, quite efficient for estimating model parameters in the identified model and also robust with respect to model misspecification. To facilitate the use of optimal design ideas in practice, we have also constructed a website that freely enables practitioners to generate a variety of optimal designs for a range of models and also enables them to evaluate the efficiency of any design.

We adapt the techniques in Stigler [Ann. Statist. 1 (1973) 472–477] to obtain a new, general asymptotic result for trimmed $U$-statistics via the generalized $L$-statistic representation introduced by Serfling [Ann. Statist. 12 (1984) 76–86]. Unlike existing results, we do not require continuity of an associated distribution at the truncation points. Our results are quite general and are expressed in terms of the quantile function associated with the distribution of the $U$-statistic summands. This approach leads to improved conditions for the asymptotic normality of these trimmed $U$-statistics.

We extend known saddlepoint tail probability approximations to multivariate cases, including multivariate conditional cases. Our approximation applies to both continuous and lattice variables, and requires the existence of a cumulant generating function. The method is applied to some examples, including a real data set from a case-control study of endometrial cancer. The method contains less terms and is easier to implement than existing methods, while showing an accuracy comparable to those methods.

Second order approximate ancillaries have evolved as the primary ingredient for recent likelihood development in statistical inference. This uses quantile functions rather than the equivalent distribution functions, and the intrinsic ancillary contour is given explicitly as the plug-in estimate of the vector quantile function. The derivation uses a Taylor expansion of the full quantile function, and the linear term gives a tangent to the observed ancillary contour. For the scalar parameter case, there is a vector field that integrates to give the ancillary contours, but for the vector case, there are multiple vector fields and the Frobenius conditions for mutual consistency may not hold. We demonstrate, however, that the conditions hold in a restricted way and that this verifies the second order ancillary contours in moderate deviations. The methodology can generate an appropriate exact ancillary when such exists or an approximate ancillary for the numerical or Monte Carlo calculation of $p$-values and confidence quantiles. Examples are given, including nonlinear regression and several enigmatic examples from the literature.

We study the problem of testing composite hypotheses versus composite alternatives, using a convex duality approach. In contrast to classical results obtained by Krafft and Witting (Z. Wahrsch. Verw. Gebiete 7 (1967) 289–302), where sufficient optimality conditions are derived via Lagrange duality, we obtain necessary and sufficient optimality conditions via Fenchel duality under compactness assumptions. This approach also differs from the methodology developed in Cvitanić and Karatzas (Bernoulli 7 (2001) 79–97).

Tackling climate change is at the top of many agendas. In this context, emission trading schemes are considered as promising tools. The regulatory framework for an emission trading scheme introduces a market for emission allowances and creates a need for risk management by appropriate financial contracts. In this work, we address logical principles underlying their valuation.

By means of white noise analysis, we prove some limit theorems for nonlinear functionals of a given Volterra process. In particular, our results apply to fractional Brownian motion (fBm) and should be compared with the classical convergence results of the 1980s due to Breuer, Dobrushin, Giraitis, Major, Surgailis and Taqqu, as well as the recent advances concerning the construction of a Lévy area for fBm due to Coutin, Qian and Unterberger.

The stochastic calculus for Gaussian processes is applied to obtain a Tanaka formula for a Volterra-type multifractional Gaussian process. The existence and regularity properties of the local time of this process are obtained by means of Berman’s Fourier analytic approach.

We study the non-equilibrium dynamics of a one-dimensional interacting particle system that is a mixture of the voter model and the exclusion process. With the process started from a finite perturbation of the ground state Heaviside configuration consisting of 1’s to the left of the origin and 0’s elsewhere, we study the relaxation time $τ$, that is, the first hitting time of the ground state configuration (up to translation). We give conditions for $τ$ to be finite and for certain moments of $τ$ to be finite or infinite, and prove a result that approaches a conjecture of Belitsky et al. (Bernoulli 7 (2001) 119–144). Ours are the first non-existence-of-moments results for $τ$ for the mixture model. Moreover, we give almost sure asymptotics for the evolution of the size of the hybrid (disordered) region. Most of our results pertain to the discrete-time setting, but several transfer to continuous-time. As well as the mixture process, some of our results also cover pure exclusion. We state several significant open problems.

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel–Riesz capacity, respectively. We apply these results to a system of stochastic wave equations in spatial dimension $k ≥ 1$ driven by a $d$-dimensional spatially homogeneous additive Gaussian noise that is white in time and colored in space.

In regression problems where covariates can be naturally grouped, the group Lasso is an attractive method for variable selection since it respects the grouping structure in the data. We study the selection and estimation properties of the group Lasso in high-dimensional settings when the number of groups exceeds the sample size. We provide sufficient conditions under which the group Lasso selects a model whose dimension is comparable with the underlying model with high probability and is estimation consistent. However, the group Lasso is, in general, not selection consistent and also tends to select groups that are not important in the model. To improve the selection results, we propose an adaptive group Lasso method which is a generalization of the adaptive Lasso and requires an initial estimator. We show that the adaptive group Lasso is consistent in group selection under certain conditions if the group Lasso is used as the initial estimator.

The concentration of empirical measures is studied for dependent data, whose joint distribution satisfies Poincaré-type or logarithmic Sobolev inequalities. The general concentration results are then applied to spectral empirical distribution functions associated with high-dimensional random matrices.