The study of non-stationary processes whose local form has controlled properties is a fruitful and important area of research, both in theory and applications. In (J. Theoret. Probab. 22 (2009) 375–401), a particular way of constructing such processes was investigated, leading in particular to multifractional multistable processes, which were built using sums over Poisson processes. We present here a different construction of these processes, based on the Ferguson–Klass–LePage series representation of stable processes. We consider various particular cases of interest, including multistable Lévy motion, multistable reverse Ornstein–Uhlenbeck process, log-fractional multistable motion and linear multistable multifractional motion. We also show that the processes defined here have the same finite dimensional distributions as the corresponding processes built in (J. Theoret. Probab. 22 (2009) 375–401). Finally, we display numerical experiments showing graphs of synthesized paths of such processes.

We derive explicitly the coupling property for the transition semigroup of a Lévy process and gradient estimates for the associated semigroup of transition operators. This is based on the asymptotic behaviour of the symbol or the characteristic exponent near zero and infinity, respectively. Our results can be applied to a large class of Lévy processes, including stable Lévy processes, layered stable processes, tempered stable processes and relativistic stable processes.

Chen, Fitzsimmons, Kuwae and Zhang (Ann. Probab. 36 (2008) 931–970) have established an Itô formula consisting in the development of $F(u(X))$ for a symmetric Markov process $X$, a function $u$ in the Dirichlet space of $X$ and any $\mathcal{C} ^{2}$-function $F$. We give here an extension of this formula for $u$ locally in the Dirichlet space of $X$ and $F$ admitting a locally bounded Radon–Nikodym derivative. This formula has some analogies with various extended Itô formulas for semi-martingales using the local time stochastic calculus. But here the part of the local time is played by a process $(\Gamma^{a}_{t},a\in \mathbb{R} ,t\geq0)$ defined thanks to Nakao’s operator (Z. Wahrsch. Verw. Gebiete 68 (1985) 557–578).

It is known that in many cases distributions of exponential integrals of Lévy processes are infinitely divisible and in some cases they are also selfdecomposable. In this paper, we give some sufficient conditions under which distributions of exponential integrals are not only selfdecomposable but furthermore are generalized gamma convolution. We also study exponential integrals of more general independent increment processes. Several examples are given for illustration.

In this paper, we study the asymptotic behavior of sums of functions of the increments of a given semimartingale, taken along a regular grid whose mesh goes to $0$. The function of the $i$th increment may depend on the current time, and also on the past of the semimartingale before this time. We study the convergence in probability of two types of such sums, and we also give associated central limit theorems. This extends known results when the summands are a function depending only on the increments, and this is motivated mainly by statistical applications.

We present an iterative sampling method which delivers upper and lower bounding processes for the Brownian path. We develop such processes with particular emphasis on being able to unbiasedly simulate them on a personal computer. The dominating processes converge almost surely in the supremum and $L_{1}$ norms. In particular, the rate of converge in $L_{1}$ is of the order $\mathcal{O}(\mathcal{K}^{-1/2})$, $\mathcal{K}$ denoting the computing cost. The a.s. enfolding of the Brownian path can be exploited in Monte Carlo applications involving Brownian paths whence our algorithm (termed the $\varepsilon$-strong algorithm) can deliver unbiased Monte Carlo estimators over path expectations, overcoming discretisation errors characterising standard approaches. We will show analytical results from applications of the $\varepsilon$-strong algorithm for estimating expectations arising in option pricing. We will also illustrate that individual steps of the algorithm can be of separate interest, giving new simulation methods for interesting Brownian distributions.

Let $\mathbf{X}=(X_{1},X_{2},X_{3})$ be a spherically symmetric random vector of which only $(X_{1},X_{2})$ can be observed. We focus attention on estimating $F$, the distribution function of the squared radius $Z:=X_{1}^{2}+X_{2}^{2}+X_{3}^{2}$, from a random sample of $(X_{1},X_{2})$. Such a problem arises in astronomy where $(X_{1},X_{2},X_{3})$ denotes the three dimensional position of a star in a galaxy but we can only observe the projected stellar positions $(X_{1},X_{2})$. We consider isotonic estimators of $F$ and derive their limit distributions. The results are nonstandard with a rate of convergence $\sqrt{n/{\log n}}$. The isotonized estimators of $F$ have exactly half the limiting variance when compared to naive estimators, which do not incorporate the shape constraint. We consider the problem of constructing point-wise confidence intervals for $F$, state sufficient conditions for the consistency of a bootstrap procedure, and show that the conditions are met by the conventional bootstrap method (generating samples from the empirical distribution function).

In Bayesian nonparametric inference, random discrete probability measures are commonly used as priors within hierarchical mixture models for density estimation and for inference on the clustering of the data. Recently, it has been shown that they can also be exploited in species sampling problems: indeed they are natural tools for modeling the random proportions of species within a population thus allowing for inference on various quantities of statistical interest. For applications that involve large samples, the exact evaluation of the corresponding estimators becomes impracticable and, therefore, asymptotic approximations are sought. In the present paper, we study the limiting behaviour of the number of new species to be observed from further sampling, conditional on observed data, assuming the observations are exchangeable and directed by a normalized generalized gamma process prior. Such an asymptotic study highlights a connection between the normalized generalized gamma process and the two-parameter Poisson–Dirichlet process that was previously known only in the unconditional case.

This paper deals with the Fisher-consistency, weak continuity and differentiability of estimating functionals corresponding to a class of both linear and nonlinear regression high breakdown M estimates, which includes S and MM estimates. A restricted type of differentiability, called weak differentiability, is defined, which suffices to prove the asymptotic normality of estimates based on the functionals. This approach allows to prove the consistency, asymptotic normality and qualitative robustness of M estimates under more general conditions than those required in standard approaches. In particular, we prove that regression MM-estimates are asymptotically normal when the observations are $\phi$-mixing.

For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik–Chervonenkis (VC) dimension or (ii) for every $\varepsilon >0$ there is a finite partition $\pi$ such the essential $\pi$-boundary of each set has measure at most $\varepsilon $. Immediate corollaries include the fact that a separable family with finite VC dimension has finite bracketing numbers, and satisfies uniform laws of large numbers for every ergodic process. From these corollaries, we derive analogous results for VC major and VC graph families of functions.

When auxiliary information is available at the design stage, samples may be selected by means of balanced sampling. Deville and Tillé proposed in 2004 a general algorithm to perform balanced sampling, named the cube method. In this paper, we are interested in a particular case of the cube method named pivotal sampling, and first described by Deville and Tillé in 1998. We show that this sampling algorithm, when applied to units ranked in a fixed order, is equivalent to Deville’s systematic sampling, in the sense that both algorithms lead to the same sampling design. This characterization enables the computation of the second-order inclusion probabilities for pivotal sampling. We show that the pivotal sampling enables to take account of an appropriate ordering of the units to achieve a variance reduction, while limiting the loss of efficiency if the ordering is not appropriate.

We prove large and moderate deviation principles for the distribution of an empirical mean conditioned by the value of the sum of discrete i.i.d. random variables. Some applications for combinatoric problems are discussed.

Consider informative selection of a sample from a finite population. Responses are realized as independent and identically distributed (i.i.d.) random variables with a probability density function (p.d.f.) $f$, referred to as the superpopulation model. The selection is informative in the sense that the sample responses, given that they were selected, are not i.i.d. $f$. In general, the informative selection mechanism may induce dependence among the selected observations. The impact of such dependence on the empirical cumulative distribution function (c.d.f.) is studied. An asymptotic framework and weak conditions on the informative selection mechanism are developed under which the (unweighted) empirical c.d.f. converges uniformly, in $L_{2}$ and almost surely, to a weighted version of the superpopulation c.d.f. This yields an analogue of the Glivenko–Cantelli theorem. A series of examples, motivated by real problems in surveys and other observational studies, shows that the conditions are verifiable for specified designs.

The paper focuses on a class of light-tailed multivariate probability distributions. These are obtained via a transformation of the margins from a heavy-tailed original distribution. This class was introduced in Balkema et al. (J. Multivariate Anal. 101 (2010) 1738–1754). As shown there, for the light-tailed meta distribution the sample clouds, properly scaled, converge onto a deterministic set. The shape of the limit set gives a good description of the relation between extreme observations in different directions. This paper investigates how sensitive the limit shape is to changes in the underlying heavy-tailed distribution. Copulas fit in well with multivariate extremes. By Galambos’s theorem, existence of directional derivatives in the upper endpoint of the copula is necessary and sufficient for convergence of the multivariate extremes provided the marginal maxima converge. The copula of the max-stable limit distribution does not depend on the margins. So margins seem to play a subsidiary role in multivariate extremes. The theory and examples presented in this paper cast a different light on the significance of margins. For light-tailed meta distributions, the asymptotic behaviour is very sensitive to perturbations of the underlying heavy-tailed original distribution, it may change drastically even when the asymptotic behaviour of the heavy-tailed density is not affected.

Let ${\mathbf{X}}_{p}=({\mathbf{s}}_{1},\ldots,{\mathbf{s}}_{n})=(X_{ij})_{p\times n}$ where $X_{ij}$’s are independent and identically distributed (i.i.d.) random variables with $EX_{11}=0$, $EX_{11}^{2}=1$ and $EX_{11}^{4}<\infty$. It is showed that the largest eigenvalue of the random matrix ${\mathbf{A}}_{p}=\frac{1}{2\sqrt{np}}({\mathbf{X}}_{p}{\mathbf{X}}_{p}^{\prime}-n{\mathbf{I}}_{p})$ tends to $1$ almost surely as $p\rightarrow\infty$, $n\rightarrow\infty$ with $p/n\rightarrow0$.

We consider the problem of testing the parametric form of the volatility for high frequency data. It is demonstrated that in the presence of microstructure noise commonly used tests do not keep the preassigned level and are inconsistent. The concept of preaveraging is used to construct new tests, which do not suffer from these drawbacks. These tests are based on a Kolmogorov–Smirnov or Cramér–von-Mises functional of an integrated stochastic process, for which weak convergence to a (conditional) Gaussian process is established. The finite sample properties of a bootstrap version of the test are illustrated by means of a simulation study.

Time-series data with regular and/or seasonal long-memory are often aggregated before analysis. Often, the aggregation scale is large enough to remove any short-memory components of the underlying process but too short to eliminate seasonal patterns of much longer periods. In this paper, we investigate the limiting correlation structure of aggregate time series within an intermediate asymptotic framework that attempts to capture the aforementioned sampling scheme. In particular, we study the autocorrelation structure and the spectral density function of aggregates from a discrete-time process. The underlying discrete-time process is assumed to be a stationary Seasonal AutoRegressive Fractionally Integrated Moving-Average (SARFIMA) process, after suitable number of differencing if necessary, and the seasonal periods of the underlying process are multiples of the aggregation size. We derive the limit of the normalized spectral density function of the aggregates, with increasing aggregation. The limiting aggregate (seasonal) long-memory model may then be useful for analyzing aggregate time-series data, which can be estimated by maximizing the Whittle likelihood. We prove that the maximum Whittle likelihood estimator (spectral maximum likelihood estimator) is consistent and asymptotically normal, and study its finite-sample properties through simulation. The efficacy of the proposed approach is illustrated by a real-life internet traffic example.