We investigate the properties of the hybrid Monte Carlo algorithm (HMC) in high dimensions. HMC develops a Markov chain reversible with respect to a given target distribution $\Pi$ using separable Hamiltonian dynamics with potential $-\log\Pi$. The additional momentum variables are chosen at random from the Boltzmann distribution, and the continuous-time Hamiltonian dynamics are then discretised using the leapfrog scheme. The induced bias is removed via a Metropolis–Hastings accept/reject rule. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an $\mathcal{O}(1)$ acceptance probability as the dimension $d$ of the state space tends to $\infty$, the leapfrog step size $h$ should be scaled as $h=l\times d^{-1/4}$. Therefore, in high dimensions, HMC requires $\mathcal{O}(d^{1/4})$ steps to traverse the state space. We also identify analytically the asymptotically optimal acceptance probability, which turns out to be $0.651$ (to three decimal places). This value optimally balances the cost of generating a proposal, which decreases as $l$ increases (because fewer steps are required to reach the desired final integration time), against the cost related to the average number of proposals required to obtain acceptance, which increases as $l$ increases.

This paper develops a framework for the estimation of the functional mean and the functional principal components when the functions form a random field. More specifically, the data we study consist of curves $X(\mathbf{s}_{k};t)$, $t\in[0,T]$, observed at spatial points $\mathbf{s}_{1},\mathbf{s}_{2},\ldots,\mathbf{s}_{N}$. We establish conditions for the sample average (in space) of the $X(\mathbf{s}_{k})$ to be a consistent estimator of the population mean function, and for the usual empirical covariance operator to be a consistent estimator of the population covariance operator. These conditions involve an interplay of the assumptions on an appropriately defined dependence between the functions $X(\mathbf{s}_{k})$ and the assumptions on the spatial distribution of the points $\mathbf{s}_{k}$. The rates of convergence may be the same as for i.i.d. functional samples, but generally depend on the strength of dependence and appropriately quantified distances between the points $\mathbf{s}_{k}$. We also formulate conditions for the lack of consistency.

In this paper, we establish relationships between four important concepts: (a) hitting time problems of Brownian motion, (b) 3-dimensional Bessel bridges, (c) Schrödinger’s equation with linear potential, and (d) heat equation problems with moving boundary. We relate (a) and (b) by means of Girsanov’s theorem, which suggests a strategy to extend our ideas to problems in $\mathbb{R}^{n}$ and general diffusions. This approach also leads to (c) because we may relate, through a Feynman–Kac representation, functionals of a Bessel bridge with two Schrödinger-type problems. In particular, we also find a fundamental solution to a class of parabolic partial differential equations with linear potential. Finally, the relationship between (c) and (d) suggests a possible link between Generalized Airy processes and their hitting times.

We consider a circular deconvolution problem, in which the density $f$ of a circular random variable $X$ must be estimated nonparametrically based on an i.i.d. sample from a noisy observation $Y$ of $X$. The additive measurement error is supposed to be independent of $X$. The objective of this work was to construct a fully data-driven estimation procedure when the error density $\varphi$ is unknown. We assume that in addition to the i.i.d. sample from $Y$, we have at our disposal an additional i.i.d. sample drawn independently from the error distribution. We first develop a minimax theory in terms of both sample sizes. We propose an orthogonal series estimator attaining the minimax rates but requiring optimal choice of a dimension parameter depending on certain characteristics of $f$ and $\varphi$, which are not known in practice. The main issue addressed in this work is the adaptive choice of this dimension parameter using a model selection approach. In a first step, we develop a penalized minimum contrast estimator assuming that the error density is known. We show that this partially adaptive estimator can attain the lower risk bound up to a constant in both sample sizes $n$ and $m$. Finally, by randomizing the penalty and the collection of models, we modify the estimator such that it no longer requires any previous knowledge of the error distribution. Even when dispensing with any hypotheses on $\varphi$, this fully data-driven estimator still preserves minimax optimality in almost the same cases as the partially adaptive estimator. We illustrate our results by computing minimal rates under classical smoothness assumptions.

This paper relates – for point processes $\Phi$ on $\mathbb{R}$ – two types of asymptotic mean stationarity (AMS) properties and several absolute continuity results for the common probability measures emerging from point process theory. It is proven that $\Phi$ is AMS under the time-shifts if and only if it is AMS under the event-shifts. The consequences for the accompanying two types of ergodic theorem are considered. Furthermore, the AMS properties are equivalent or closely related to several absolute continuity results. Thus, the class of AMS point processes is characterized in several ways. Many results from stationary point process theory are generalized for AMS point processes. To obtain these results, we first use Campbell’s equation to rewrite the well-known Palm relationship for general nonstationary point processes into expressions which resemble results from stationary point process theory.

Since the seminal work by Nagel and Weiss, the iteration stable (STIT) tessellations have attracted considerable interest in stochastic geometry as a natural and flexible, yet analytically tractable model for hierarchical spatial cell-splitting and crack-formation processes. We provide in this paper a fundamental link between typical characteristics of STIT tessellations and those of suitable mixtures of Poisson hyperplane tessellations using martingale techniques and general theory of piecewise deterministic Markov processes (PDMPs). As applications, new mean values and new distributional results for the STIT model are obtained.

For the problem of estimating lower tail and upper tail copulas, we propose two bootstrap procedures for approximating the distribution of the corresponding empirical tail copulas. The first method uses a multiplier bootstrap of the empirical tail copula process and requires estimation of the partial derivatives of the tail copula. The second method avoids this estimation problem and uses multipliers in the two-dimensional empirical distribution function and in the estimates of the marginal distributions. For both multiplier bootstrap procedures, we prove consistency.

For these investigations, we demonstrate that the common assumption of the existence of continuous partial derivatives in the the literature on tail copula estimation is so restrictive, such that the tail copula corresponding to tail independence is the only tail copula with this property. Moreover, we are able to solve this problem and prove weak convergence of the empirical tail copula process under nonrestrictive smoothness assumptions that are satisfied for many commonly used models. These results are applied in several statistical problems, including minimum distance estimation and goodness-of-fit testing.

In this paper we consider a heavy-tailed stochastic volatility model, $X_{t}=\sigma_{t}Z_{t}$, $t\in\mathbb{Z}$, where the volatility sequence $(\sigma_{t})$ and the i.i.d. noise sequence $(Z_{t})$ are assumed independent, $(\sigma_{t})$ is regularly varying with index $\alpha>0$, and the $Z_{t}$’s have moments of order larger than $\alpha$. In the literature (see Ann. Appl. Probab. 8 (1998) 664–675, J. Appl. Probab. 38A (2001) 93–104, In Handbook of Financial Time Series (2009) 355–364 Springer), it is typically assumed that $(\log\sigma_{t})$ is a Gaussian stationary sequence and the $Z_{t}$’s are regularly varying with some index $\alpha$ (i.e., $(\sigma_{t})$ has lighter tails than the $Z_{t}$’s), or that $(Z_{t})$ is i.i.d. centered Gaussian. In these cases, we see that the sequence $(X_{t})$ does not exhibit extremal clustering. In contrast to this situation, under the conditions of this paper, both situations are possible; $(X_{t})$ may or may not have extremal clustering, depending on the clustering behavior of the $\sigma$-sequence.

In this paper we propose using a nonparametric model specification test for parametric time series with long-range dependence (LRD). To establish asymptotic distributions of the proposed test statistic, we develop new central limit theorems for certain weighted quadratic forms of stationary time series with LRD. To implement our proposed test in practice, we develop a computer-intensive parametric bootstrap simulation procedure for finding simulated critical values. As a result, our finite-sample studies demonstrate that both the proposed theory and the simulation procedure work well, and that the proposed test has little size distortion and reasonable power.

We study the problem of approximation of solutions of the Skorokhod problem and reflecting stochastic differential equations (SDEs) with jumps by sequences of solutions of equations with penalization terms. Applications to discrete approximation of weak and strong solutions of reflecting SDEs are given. Our proofs are based on new estimates for solutions of equations with penalization terms and the theory of convergence in the Jakubowski $S$-topology.

Linear combinations of independent random variables have been extensively studied in the literature. However, most of the work is based on some specific distribution assumptions. In this paper, a companion of (J. Appl. Probab. 48 (2011) 1179–1188), we unify the study of linear combinations of independent nonnegative random variables under the general setup by using some monotone transforms. The results are further generalized to the case of independent but not necessarily identically distributed nonnegative random variables. The main results complement and generalize the results in the literature including (In Studies in Econometrics, Time Series, and Multivariate Statistics (1983) 465–489 Academic Press; Sankhyā Ser. A 60 (1998) 171–175; Sankhyā Ser. A 63 (2001) 128–132; J. Statist. Plann. Inference 92 (2001) 1–5; Bernoulli 17 (2011) 1044–1053).

Cognitive assessment is a growing area in psychological and educational measurement, where tests are given to assess mastery/deficiency of attributes or skills. A key issue is the correct identification of attributes associated with items in a test. In this paper, we set up a mathematical framework under which theoretical properties may be discussed. We establish sufficient conditions to ensure that the attributes required by each item are learnable from the data.

We display several examples of generalized gamma convoluted and hyperbolically completely monotone random variables related to positive $\alpha$-stable laws. We also obtain new factorizations for the latter, refining Kanter’s and Pestana–Shanbhag–Sreehari’s. These results give stronger credit to Bondesson’s hypothesis that positive $\alpha$-stable densities are hyperbolically completely monotone whenever $\alpha\le1/2$.

We study the least squares estimator in the residual variance estimation context. We show that the mean squared differences of paired observations are asymptotically normally distributed. We further establish that, by regressing the mean squared differences of these paired observations on the squared distances between paired covariates via a simple least squares procedure, the resulting variance estimator is not only asymptotically normal and root-$n$ consistent, but also reaches the optimal bound in terms of estimation variance. We also demonstrate the advantage of the least squares estimator in comparison with existing methods in terms of the second order asymptotic properties.

Given a birth–death process on $\mathbb{N} $ with semigroup $(P_{t})_{t\geq0}$ and a discrete gradient ${\partial}_{u}$ depending on a positive weight $u$, we establish intertwining relations of the form ${\partial}_{u}P_{t}=Q_{t}\,{\partial}_{u}$, where $(Q_{t})_{t\geq0}$ is the Feynman–Kac semigroup with potential $V_{u}$ of another birth–death process. We provide applications when $V_{u}$ is nonnegative and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. Our analysis is naturally connected to the previous works of Caputo–Dai Pra–Posta and of Chen on birth–death processes. The proofs are remarkably simple and rely on interpolation, commutation, and convexity.

Consider on a manifold the solution $X$ of a stochastic differential equation driven by a Lévy process without Brownian part. Sufficient conditions for the smoothness of the law of $X_{t}$ are given, with particular emphasis on noncompact manifolds. The result is deduced from the case of affine spaces by means of a localisation technique. The particular cases of Lie groups and homogeneous spaces are discussed.

Conditions are presented for different types of identifiability of discrete variable models generated over an undirected graph in which one node represents a binary hidden variable. These models can be seen as extensions of the latent class model to allow for conditional associations between the observable random variables. Since local identification corresponds to full rank of the parametrization map, we establish a necessary and sufficient condition for the rank to be full everywhere in the parameter space. The condition is based on the topology of the undirected graph associated to the model. For non-full rank models, the obtained characterization allows us to find the subset of the parameter space where the identifiability breaks down.

In this paper, we study the existence of the density associated with the exponential functional of the Lévy process $\xi$,

\[I_{\mathbf{e} _{q}}:=\int_{0}^{\mathbf{e} _{q}}\mathrm{e}^{\xi_{s}}\,\mathrm{d}s,\]

where $\mathbf{e} _{q}$ is an independent exponential r.v. with parameter $q\geq0$. In the case where $\xi$ is the negative of a subordinator, we prove that the density of $I_{\mathbf{e}_{q}}$, here denoted by $k$, satisfies an integral equation that generalizes that reported by Carmona et al. [7]. Finally, when $q=0$, we describe explicitly the asymptotic behavior at $0$ of the density $k$ when $\xi$ is the negative of a subordinator and at $\infty$ when $\xi$ is a spectrally positive Lévy process that drifts to $+\infty$.

Two test statistics are introduced to test the null hypotheses that the sampling distribution has an increasing hazard rate on a specified interval $[0,a]$. These statistics are empirical $L_{1}$-type distances between the isotonic estimates, which use the monotonicity constraint, and either the empirical distribution function or the empirical cumulative hazard. They measure the excursions of the empirical estimates with respect to the isotonic estimates, owing to local non-monotonicity. Asymptotic normality of the test statistics, if the hazard is strictly increasing on $[0,a]$, is established under mild conditions. This is done by first approximating the global empirical distance by a distance with respect to the underlying distribution function. The resulting integral is treated as sum of increasingly many local integrals to which a central limit theorem can be applied. The behavior of the local integrals is determined by a canonical process, the difference between the stochastic process $x\mapsto W(x)+x^{2}$, where $W$ is standard two-sided Brownian motion, and its greatest convex minorant.

We obtain some integrability properties and some limit theorems for the exit time from a cone of a planar Brownian motion, and we check that our computations are correct via Bougerol’s identity.

We introduce the probabilistic symbol for the class of homogeneous diffusions with jumps (in the sense of Jacod/Shiryaev). This concept generalizes the well-known characteristic exponent of a Lévy process. Using the symbol, we introduce eight indices which generalize the Blumenthal–Getoor index $\beta$ and the Pruitt index $\delta$. These indices are used afterwards to obtain growth and Hölder conditions of the process. In the future, the technical main results will be used to derive further fine properties. Since virtually all examples of homogeneous diffusions in the literature are Markovian, we construct a process which does not have this property.

We address the problem of upper bounding the mean square error of MCMC estimators. Our analysis is nonasymptotic. We first establish a general result valid for essentially all ergodic Markov chains encountered in Bayesian computation and a possibly unbounded target function $f$. The bound is sharp in the sense that the leading term is exactly $\sigma_{\mathrm{as}}^{2}(P,f)/n$, where $\sigma_{\mathrm{as}}^{2}(P,f)$ is the CLT asymptotic variance. Next, we proceed to specific additional assumptions and give explicit computable bounds for geometrically and polynomially ergodic Markov chains under quantitative drift conditions. As a corollary, we provide results on confidence estimation.

When collections of functional data are too large to be exhaustively observed, survey sampling techniques provide an effective way to estimate global quantities such as the population mean function. Assuming functional data are collected from a finite population according to a probabilistic sampling scheme, with the measurements being discrete in time and noisy, we propose to first smooth the sampled trajectories with local polynomials and then estimate the mean function with a Horvitz–Thompson estimator. Under mild conditions on the population size, observation times, regularity of the trajectories, sampling scheme, and smoothing bandwidth, we prove a Central Limit theorem in the space of continuous functions. We also establish the uniform consistency of a covariance function estimator and apply the former results to build confidence bands for the mean function. The bands attain nominal coverage and are obtained through Gaussian process simulations conditional on the estimated covariance function. To select the bandwidth, we propose a cross-validation method that accounts for the sampling weights. A simulation study assesses the performance of our approach and highlights the influence of the sampling scheme and bandwidth choice.

Consider a first-order autoregressive process $X_{i}=\beta X_{i-1}+\varepsilon_{i}$, where $\varepsilon_{i}=G(\eta_{i},\eta_{i-1},\ldots)$ and $\eta_{i}$, $i\in\mathbb{Z}$ are i.i.d. random variables. Motivated by two important issues for the inference of this model, namely, the quantile inference for $H_{0}\colon\ \beta=1$, and the goodness-of-fit for the unit root model, the notion of the marked empirical process $\alpha_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}g(X_{i}/a_{n})I(\varepsilon_{i}\leq x)$, $x\in\mathbb{R}$ is investigated in this paper. Herein, $g(\cdot)$ is a continuous function on $\mathbb{R}$ and $\{a_{n}\}$ is a sequence of self-normalizing constants. As the innovation $\{\varepsilon_{i}\}$ is usually not observable, the residual marked empirical process $\hat{\alpha}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}g(X_{i}/a_{n})I(\hat{\varepsilon}_{i}\leq x)$, $x\in\mathbb{R}$, is considered instead, where $\hat{\varepsilon}_{i}=X_{i}-\hat{\beta}X_{i-1}$ and $\hat{\beta}$ is a consistent estimate of $\beta$. In particular, via the martingale decomposition of stationary process and the stochastic integral result of Jakubowski (Ann. Probab. 24 (1996) 2141–2153), the limit distributions of $\alpha_{n}(x)$ and $\hat{\alpha}_{n}(x)$ are established when $\{\varepsilon_{i}\}$ is a short-memory process. Furthermore, by virtue of the results of Wu (Bernoulli 95 (2003) 809–831) and Ho and Hsing (Ann. Statist. 24 (1996) 992–1024) of empirical process and the integral result of Mikosch and Norvaiša (Bernoulli 6 (2000) 401–434) and Young (Acta Math. 67 (1936) 251–282), the limit distributions of $\alpha_{n}(x)$ and $\hat{\alpha}_{n}(x)$ are also derived when $\{\varepsilon_{i}\}$ is a long-memory process.

We consider a quadratic functional regression model in which a scalar response depends on a functional predictor; the common functional linear model is a special case. We wish to test the significance of the nonlinear term in the model. We develop a testing method which is based on projecting the observations onto a suitably chosen finite dimensional space using functional principal component analysis. The asymptotic behavior of our testing procedure is established. A simulation study shows that the testing procedure has good size and power with finite sample sizes. We then apply our test to a data set provided by Tecator, which consists of near-infrared absorbance spectra and fat content of meat.