Operator fractional Brownian motions (OFBMs) are (i) Gaussian, (ii) operator self-similar and (iii) stationary increment processes. They are the natural multivariate generalizations of the well-studied fractional Brownian motions. Because of the possible lack of time-reversibility, the defining properties (i)–(iii) do not, in general, characterize the covariance structure of OFBMs. To circumvent this problem, the class of OFBMs is characterized here by means of their integral representations in the spectral and time domains. For the spectral domain representations, this involves showing how the operator self-similarity shapes the spectral density in the general representation of stationary increment processes. The time domain representations are derived by using primary matrix functions and taking the Fourier transforms of the deterministic spectral domain kernels. Necessary and sufficient conditions for OFBMs to be time-reversible are established in terms of their spectral and time domain representations. It is also shown that the spectral density of the stationary increments of an OFBM has a rigid structure, here called the dichotomy principle. The notion of operator Brownian motions is also explored.

In this paper, we introduce a new class of Lévy processes which we call hypergeometric-stable Lévy processes because they are obtained from symmetric stable processes through several transformations, where the Gauss hypergeometric function plays an essential role. We characterize the Lévy measure of this class and obtain several useful properties such as the Wiener–Hopf factorization, the characteristic exponent and some associated exit problems.

We present a general principle for estimating a regression function nonparametrically, allowing for a wide variety of data filtering, for example, repeated left truncation and right censoring. Both the mean and the median regression cases are considered. The method works by first estimating the conditional hazard function or conditional survivor function and then integrating. We also investigate improved methods that take account of model structure such as independent errors and show that such methods can improve performance when the model structure is true. We establish the pointwise asymptotic normality of our estimators.

In this paper, we prove maximal inequalities and study the functional central limit theorem for the partial sums of linear processes generated by dependent innovations. Due to the general weights, these processes can exhibit long-range dependence and the limiting distribution is a fractional Brownian motion. The proofs are based on new approximations by a linear process with martingale difference innovations. The results are then applied to study an estimator of the isotonic regression when the error process is a (possibly long-range dependent) time series.

In this paper, we introduce and examine a fractional linear birth–death process $N_ν(t), t>0$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^ν(t), t>0, k≥0$. We present a subordination relationship connecting $N_ν(t), t>0$, with the classical birth–death process $N(t), t>0$, by means of the time process $T_{2ν}(t), t>0$, whose distribution is related to a time-fractional diffusion equation.

We obtain explicit formulas for the extinction probability $p_0^ν(t)$ and the state probabilities $p_k^ν(t), t>0, k≥1$, in the three relevant cases $λ>μ, λ<μ, λ=μ$ (where $λ$ and $μ$ are, respectively, the birth and death rates) and discuss their behaviour in specific situations. We highlight the connection of the fractional linear birth–death process with the fractional pure birth process. Finally, the mean values $\mathbb{E}N_{\nu}(t)$ and $\operatorname{\mathbb{V}ar}N_{\nu}(t)$ are derived and analyzed.

This paper studies the long-time behavior of the empirical distribution of age and normalized position of an age-dependent supercritical branching Markov process. The motion of each individual during its life is a random function of its age. It is shown that the empirical distribution of the age and the normalized position of all individuals alive at time $t$ converges as $t→∞$ to a deterministic product measure.

We establish an optimal transportation inequality for the Poisson measure on the configuration space. Furthermore, under the Dobrushin uniqueness condition, we obtain a sharp transportation inequality for the Gibbs measure on $ℕ^Λ$ or the continuum Gibbs measure on the configuration space.

We use Rice formulae in order to compute the moments of some level functionals which are linked to problems in oceanography and optics: the number of specular points in one and two dimensions, the distribution of the normal angle of level curves and the number of dislocations in random wavefronts. We compute expectations and, in some cases, also second moments of such functionals. Moments of order greater than one are more involved, but one needs them whenever one wants to perform statistical inference on some parameters in the model or to test the model itself. In some cases, we are able to use these computations to obtain a central limit theorem.

Let $\{X(t) :t∈[0, ∞)\}$ be a centered Gaussian process with stationary increments and variance function $σ_X^2(t)$. We study the exact asymptotics of $ℙ(\sup _{t∈[0, T]}X(t)>u)$ as $u→∞$, where $T$ is an independent of $\{X(t)\}$ non-negative Weibullian random variable. As an illustration, we work out the asymptotics of the supremum distribution of fractional Laplace motion.

The so-called pinball loss for estimating conditional quantiles is a well-known tool in both statistics and machine learning. So far, however, only little work has been done to quantify the efficiency of this tool for nonparametric approaches. We fill this gap by establishing inequalities that describe how close approximate pinball risk minimizers are to the corresponding conditional quantile. These inequalities, which hold under mild assumptions on the data-generating distribution, are then used to establish so-called variance bounds, which recently turned out to play an important role in the statistical analysis of (regularized) empirical risk minimization approaches. Finally, we use both types of inequalities to establish an oracle inequality for support vector machines that use the pinball loss. The resulting learning rates are min–max optimal under some standard regularity assumptions on the conditional quantile.

Multivariate extreme value theory assumes a multivariate domain of attraction condition for the distribution of a random vector. This necessitates that each component satisfies a marginal domain of attraction condition. An approximation of the joint distribution of a random vector obtained by conditioning on one of the components being extreme was developed by Heffernan and Tawn [12] and further studied by Heffernan and Resnick [11]. These papers left unresolved the consistency of different models obtained by conditioning on different components being extreme and we here provide clarification of this issue. We also clarify the relationship between these conditional distributions, multivariate extreme value theory and standard regular variation on cones of the form $[0, ∞]×(0, ∞]$.

It is often reasonable to assume that the dependence structure of a bivariate continuous distribution belongs to the class of extreme-value copulas. The latter are characterized by their Pickands dependence function. In this paper, a procedure is proposed for testing whether this function belongs to a given parametric family. The test is based on a Cramér–von Mises statistic measuring the distance between an estimate of the parametric Pickands dependence function and either one of two nonparametric estimators thereof studied by Genest and Segers [Ann. Statist. 37 (2009) 2990–3022]. As the limiting distribution of the test statistic depends on unknown parameters, it must be estimated via a parametric bootstrap procedure, the validity of which is established. Monte Carlo simulations are used to assess the power of the test and an extension to dependence structures that are left-tail decreasing in both variables is considered.

Let $\{X_i\}_{i≥1}$ be an i.i.d. sequence of random variables and define, for $n≥2$, $$T_{n}=\cases{n^{-1/2}\hat{\sigma}_{n}^{-1}S_{n}, & $\quad \hat{\sigma}_{n}>0,$ \cr 0, & $\quad \hat{\sigma}_{n}=0,$} \qquad\mbox{with }S_{n}=\sum_{i=1}^{n}X_{i}, \hat{\sigma}^{2}_{n}=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-n^{-1}S_{n})^{2}.$$ We investigate the connection between the distribution of an observation $X_i$ and finiteness of $\mathrm{E}|T_n|^r$ for $(n,r)∈ℕ_{≥2}×ℝ^+$. Moreover, assuming $T_{n}\stackrel {d}{\longrightarrow }T$, we prove that for any $r>0, \lim _{n→∞}\mathrm{E}|T_n|^r=\mathrm{E}|T|^r<∞$, provided there is an integer $n_0$ such that $\mathrm{E}|T_{n_0}|^r$ is finite.

The aim of this article is to establish asymptotic distributions and consistency of subsampling for spectral density and for magnitude of coherence for non-stationary, almost periodically correlated time series. We show the asymptotic normality of the spectral density estimator and the limiting distribution of a magnitude of coherence statistic for all points from the bifrequency square. The theoretical results hold under $α$-mixing and moment conditions.

There exist very few results on mixing for non-stationary processes. However, mixing is often required in statistical inference for non-stationary processes such as time-varying ARCH (tvARCH) models. In this paper, bounds for the mixing rates of a stochastic process are derived in terms of the conditional densities of the process. These bounds are used to obtain the $α$, 2-mixing and $β$-mixing rates of the non-stationary time-varying ARCH($p$) process and ARCH($∞$) process. It is shown that the mixing rate of the time-varying ARCH($p$) process is geometric, whereas the bound on the mixing rate of the ARCH($∞$) process depends on the rate of decay of the ARCH($∞$) parameters. We note that the methodology given in this paper is applicable to other processes.

This article considers the problem of multiple hypothesis testing using $t$-tests. The observed data are assumed to be independently generated conditional on an underlying and unknown two-state hidden model. We propose an asymptotically valid data-driven procedure to find critical values for rejection regions controlling the $k$-familywise error rate ($k$-FWER), false discovery rate (FDR) and the tail probability of false discovery proportion (FDTP) by using one-sample and two-sample $t$-statistics. We only require a finite fourth moment plus some very general conditions on the mean and variance of the population by virtue of the moderate deviations properties of $t$-statistics. A new consistent estimator for the proportion of alternative hypotheses is developed. Simulation studies support our theoretical results and demonstrate that the power of a multiple testing procedure can be substantially improved by using critical values directly, as opposed to the conventional $p$-value approach. Our method is applied in an analysis of the microarray data from a leukemia cancer study that involves testing a large number of hypotheses simultaneously.

We explore statistical inference in self-similar conservative fragmentation chains when only approximate observations of the sizes of the fragments below a given threshold are available. This framework, introduced by Bertoin and Martinez [Adv. Appl. Probab. 37 (2005) 553–570], is motivated by mineral crushing in the mining industry. The underlying object that can be identified from the data is the step distribution of the random walk associated with a randomly tagged fragment that evolves along the genealogical tree representation of the fragmentation process. We compute upper and lower rates of estimation in a parametric framework and show that in the nonparametric case, the difficulty of the estimation is comparable to ill-posed linear inverse problems of order 1 in signal denoising.

In this paper, we develop necessary and sufficient conditions for the validity of a martingale approximation for the partial sums of a stationary process in terms of the maximum of consecutive errors. Such an approximation is useful for transferring the conditional functional central limit theorem from the martingale to the original process. The condition found is simple and well adapted to a variety of examples, leading to a better understanding of the structure of several stochastic processes and their asymptotic behaviors. The approximation brings together many disparate examples in probability theory. It is valid for classes of variables defined by familiar projection conditions such as the Maxwell–Woodroofe condition, various classes of mixing processes, including the large class of strongly mixing processes, and for additive functionals of Markov chains with normal or symmetric Markov operators.

We show that a large subclass of variograms is closed under products and that some desirable stability properties, such as the product of special compositions, can be obtained within the proposed setting. We introduce new classes of kernels of Schoenberg–Lévy type and demonstrate some important properties of rotationally invariant variograms.

We characterize the Gneiting class of space–time covariance functions and give more relaxed conditions on the functions involved. We then show necessary conditions for the construction of compactly supported functions of the Gneiting type. These conditions are very general since they do not depend on the Euclidean norm.

We provide a representation of the maximal difference between a standard Brownian bridge and its concave majorant on the unit interval, from which we deduce expressions for the distribution and density functions and moments of this difference. This maximal difference has an application in nonparametric statistics, where it arises in testing monotonicity of a density or regression curve.

Using Riemann–Stieltjes methods for integrators of bounded $p$-variation we define a pathwise integral driven by a fractional Lévy process (FLP). To explicitly solve general fractional stochastic differential equations (SDEs) we introduce an Ornstein–Uhlenbeck model by a stochastic integral representation, where the driving stochastic process is an FLP. To achieve the convergence of improper integrals, the long-time behavior of FLPs is derived. This is sufficient to define the fractional Lévy–Ornstein–Uhlenbeck process (FLOUP) pathwise as an improper Riemann–Stieltjes integral. We show further that the FLOUP is the unique stationary solution of the corresponding Langevin equation. Furthermore, we calculate the autocovariance function and prove that its increments exhibit long-range dependence. Exploiting the Langevin equation, we consider SDEs driven by FLPs of bounded $p$-variation for $p<2$ and construct solutions using the corresponding FLOUP. Finally, we consider examples of such SDEs, including various state space transforms of the FLOUP and also fractional Lévy-driven Cox–Ingersoll–Ross (CIR) models.