Fitting sparse models to high-dimensional time series is an important area of statistical inference. In this paper, we consider sparse vector autoregressive models and develop appropriate bootstrap methods to infer properties of such processes. Our bootstrap methodology generates pseudo time series using a model-based bootstrap procedure which involves an estimated, sparsified version of the underlying vector autoregressive model. Inference is performed using so-called de-sparsified or de-biased estimators of the autoregressive model parameters. We derive the asymptotic distribution of such estimators in the time series context and establish asymptotic validity of the bootstrap procedure proposed for estimation and, appropriately modified, for testing purposes. In particular, we focus on testing that large groups of autoregressive coefficients equal zero. Our theoretical results are complemented by simulations which investigate the finite sample performance of the bootstrap methodology proposed. A real-life data application is also presented.

We consider hypotheses testing problems for three parameters in high-dimensional linear models with minimal sparsity assumptions of their type but without any compatibility conditions. Under this framework, we construct the first $\sqrt{\mathit{n}}$-consistent estimators for low-dimensional coefficients, the signal strength, and the noise level. We support our results using numerical simulations and provide comparisons with other estimators.

We establish uniqueness for a class of first-order Hamilton–Jacobi equations with Hamiltonians that arise from the large deviations of the empirical measure and empirical flux pair of weakly interacting Markov jump processes. As a corollary, we obtain such a large deviation principle in the context of weakly interacting processes with time-periodic rates in which the period-length converges to 0.

We consider Ewens random permutations of length n conditioned to have no cycle longer than ${\mathit{n}}^{\mathit{\beta }}$ with $0<\mathit{\beta }<1$ and study the asymptotic behaviour as $\mathit{n}\to \infty$. We obtain very precise information on the joint distribution of the lengths of the longest cycles; in particular we prove a functional limit theorem where the cumulative number of long cycles converges to a Poisson process in the suitable scaling. Furthermore, we prove convergence of the total variation distance between joint cycle counts and suitable independent Poisson random variables up to a significantly larger maximal cycle length than previously known. Finally, we remove a superfluous assumption from a central limit theorem for the total number of cycles proved in an earlier paper.

We introduce a method to prove metastability of the contact process on Erdős–Rényi graphs and on configuration model graphs. The method relies on uniformly bounding the total infection rate from below, over all sets with a fixed number of nodes. Once this bound is established, a simple comparison with a well chosen birth-and-death process will show the exponential growth of the extinction time. Our paper complements recent results on the metastability of the contact process: under a certain minimal edge density condition, we give explicit lower bounds on the infection rate needed to get metastability, and we have explicit exponentially growing lower bounds on the expected extinction time.

We provide existence, uniqueness and stability results for affine stochastic Volterra equations with ${\mathit{L}}^1$-kernels and jumps. Such equations arise as scaling limits of branching processes in population genetics and self-exciting Hawkes processes in mathematical finance. The strategy we adopt for the existence part is based on approximations using stochastic Volterra equations with ${\mathit{L}}^2$-kernels combined with a general stability result. Most importantly, we establish weak uniqueness using a duality argument on the Fourier–Laplace transform via a deterministic Riccati–Volterra integral equation. We illustrate the applicability of our results on Hawkes processes and a class of hyper-rough Volterra Heston models with a Hurst index $\mathit{H}\in (-1/2,1/2]$.

We consider a family of infinite dimensional product measures with tails between Gaussian and exponential, which we call p-exponential measures. We study their measure-theoretic properties and in particular their concentration. Our findings are used to develop a general contraction theory of posterior distributions on nonparametric models with p-exponential priors in separable Banach parameter spaces. Our approach builds on the general contraction theory for Gaussian process priors in (Ann. Statist. 36 (2008) 1435–1463), namely we use prior concentration to verify prior mass and entropy conditions sufficient for posterior contraction. However, the specific concentration properties of p-exponential priors lead to a more complex entropy bound which can influence negatively the obtained rate of contraction, depending on the topology of the parameter space. Subject to the more complex entropy bound, we show that the rate of contraction depends on the position of the true parameter relative to a certain Banach space associated to p-exponential measures and on the small ball probabilities of these measures. For example, we apply our theory in the white noise model under Besov regularity of the truth and obtain minimax rates of contraction using (rescaled) α-regular p-exponential priors. In particular, our results suggest that when interested in spatially inhomogeneous unknown functions, in terms of posterior contraction, it is preferable to use Laplace rather than Gaussian priors.

A Galton–Watson process in varying environment is a discrete time branching process where the offspring distributions vary among generations. Based on a two-spine decomposition technique, we provide a probabilistic argument of a Yaglom-type limit for this family processes. The result states that, in the critical case, a suitable normalisation of the process conditioned on non-extinction converges in distribution to a standard exponential random variable.

We will consider multivariate stochastic processes indexed either by vertices or pairs of vertices of a dynamic network. Under a dynamic network, we understand a network with a fixed vertex set and an edge set which changes randomly over time. We will assume that the spatial dependence-structure of the processes conditional on the network behaves in the following way: Close vertices (or pairs of vertices) are dependent, while we assume that the dependence decreases conditionally on that the distance in the network increases. We make this intuition mathematically precise by considering three concepts based on correlation, β-mixing with time-varying β-coefficients and conditional independence. These concepts allow proving weak-dependence results, for example, an exponential inequality, which might be of independent interest. In order to demonstrate the use of these concepts in an application, we study the asymptotics (for growing networks) of a goodness of fit test in a dynamic interaction network model based on a Cox-type model for counting processes. This model is then applied to bike-sharing data.

We study the law of the iterated logarithm (Khinchin (1924), Kolmogorov (1929)) and related strong invariance principles for functionals in stochastic geometry. As potential applications, we think of well-known functionals defined on the k-nearest neighbors graph and important functionals in topological data analysis such as the Euler characteristic and persistent Betti numbers.

This paper deduces exponential matrix concentration from a Poincaré inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on the subadditivity of Poincaré inequalities and a chain rule inequality for the trace of the matrix Dirichlet form. It also uses a symmetrization technique to avoid difficulties associated with a direct extension of the classic scalar argument.

We investigate a non-Markovian analogue of the Harris contact process in a finite connected graph $\mathit{G}=(\mathit{V},\mathit{E})$: an individual is attached to each site $\mathit{x}\in \mathit{V}$, and it can be infected or healthy; the infection propagates to healthy neighbors just as in the usual contact process, according to independent exponential times with a fixed rate $\mathit{\lambda }>0$; however, the recovery times for an individual are given by the points of a renewal process attached to its timeline, whose waiting times have distribution μ such that $\mathit{\mu }(\mathit{t},\infty )={\mathit{t}}^{-\mathit{\alpha }}\mathit{L}(\mathit{t})$, where $1/2<\mathit{\alpha }<1$ and $\mathit{L}(·)$ is a slowly varying function; the renewal processes are assumed to be independent for different sites. We show that, starting with a single infected individual, if $|\mathit{V}|<2+(2\mathit{\alpha }-1)/[(1-\mathit{\alpha })(2-\mathit{\alpha })]$, then the infection does not survive for any λ; and if $|\mathit{V}|>1/(1-\mathit{\alpha })$, then, for every λ, the infection has positive probability to survive.

Consider a moving average process X of the form $\mathit{X}(\mathit{t})={\int }_{-\infty }^{\mathit{t}}\mathit{\phi }(\mathit{t}-\mathit{u})\mathit{d}{\mathit{Z}}_{\mathit{u}}$, $\mathit{t}\ge 0$, where Z is a (non Gaussian) Hermite process of order $\mathit{q}\ge 2$ and $\mathit{\phi }:{\mathbb{R}}_{+}\to \mathbb{R}$ is sufficiently integrable. This paper investigates the fluctuations, as $\mathit{T}\to \infty$, of integral functionals of the form $\mathit{t}\mapsto {\int }_0^{\mathit{T}\mathit{t}}\mathit{P}(\mathit{X}(\mathit{s}))\mathit{d}\mathit{s}$, in the case where P is any given polynomial function. It extends a study initiated in (Stoch. Dyn. 18 (2018) 1850028, 18), where only the quadratic case $\mathit{P}(\mathit{x})={\mathit{x}}^2$ and the convergence in the sense of finite-dimensional distributions were considered.

We give a description of invariants and attractors of the critical and subcritical Galton–Watson tree measures under the operation of Horton pruning (cutting tree leaves with subsequent series reduction). Under a regularity condition, the class of invariant measures consists of the critical binary Galton–Watson tree and a one-parameter family of critical Galton–Watson trees with offspring distribution $\{{\mathit{q}}_{\mathit{k}}\}$ that has a power tail ${\mathit{q}}_{\mathit{k}}\sim \mathit{C}{\mathit{k}}^{-(1+1/{\mathit{q}}_0)}$, where ${\mathit{q}}_0\in (1/2,1)$. Each invariant measure has a non-empty domain of attraction under consecutive Horton pruning, specified by the tail behavior of the initial Galton–Watson offspring distribution. The invariant measures satisfy the Toeplitz property for the Tokunaga coefficients and obey the Horton law with exponent $\mathit{R}={(1-{\mathit{q}}_0)}^{-1/{\mathit{q}}_0}$.

A coupling method is developed for univariate extreme value theory, providing an alternative to the use of the tail empirical/quantile processes. We emphasize the Peak-over-Threshold approach that approximates the distribution above high threshold by the Generalized Pareto Distribution (GPD) and compare the empirical distribution of exceedances to the empirical distribution associated with the limit GPD model. Sharp bounds for their Wasserstein distance in the second order Wasserstein space are provided. As an application, we recover standard results on the asymptotic behavior of the Hill estimator, the Weissman extreme quantile estimator or the probability weighted moment estimators, shedding some new light on the theory.

We revisit the problem of maximising the expected length of increasing subsequence that can be selected from a marked Poisson process by an online strategy. Resorting to a natural size variable, we represent the problem in terms of a controlled piecewise deterministic Markov process with decreasing paths. We apply a comparison method to the optimality equation to obtain fairly complete asymptotic expansions for the moments of the maximal length, and, with the aid of a renewal approximation, give a novel proof to the central limit theorem for the length of selected subsequence under either the optimal strategy or a strategy sufficiently close to optimality.

A new probabilistic insight into the structure of local time is presented. A convolution formula for the local time at 0 of Itô diffusions reflecting at 0 is obtained. A simple integro-differential equation for the cumulative distribution function of the local time is given. Finally, a probabilistic representation of a generalized Stroock–Williams equation is presented.

Stochastic partial differential equations (SPDE) on graphs were recently introduced by Cerrai and Freidlin (Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 865–899). This class of stochastic equations in infinite dimensions provides a minimal framework for the study of the effective dynamics of much more complex systems. However, how they emerge from microscopic individual-based models is still poorly understood, partly due to complications near vertex singularities. In this work, motivated by the study of the dynamics and the genealogies of expanding populations in spatially structured environments, we obtain a new class of SPDE on graphs of Wright–Fisher type which have nontrivial boundary conditions on the vertex set. We show that these SPDE arise as scaling limits of suitably defined biased voter models (BVM), which extends the scaling limits of Durrett and Fan (Ann. Appl. Probab. 26 (2016) 3456–3490). We further obtain a convergent simulation scheme for each of these SPDE in terms of a system of Itô SDEs, which is useful when the size of the BVM is too large for stochastic simulations. These give the first rigorous connection between SPDE on graphs and more discrete models, specifically, interacting particle systems and interacting SDEs. Uniform heat kernel estimates for symmetric random walks approximating diffusions on graphs are the keys to our proofs. Some open problems are provided as further motivations of our study.

We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback–Leibler (KL) divergence as the objective functional. We show that an underdamped form of the Langevin algorithm performs accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm.

We consider different types of predictive intervals and ask whether they are elicitable, that is, are unique minimizers of a loss or scoring function in expectation. The equal-tailed interval is elicitable, with a rich class of suitable loss functions, though subject to translation invariance, or positive homogeneity and differentiability, the Winkler interval score becomes a unique choice. The modal interval also is elicitable, with a sole consistent scoring function, up to equivalence. However, the shortest interval fails to be elicitable relative to practically relevant classes of distributions. These results provide guidance in interval forecast evaluation and support recent choices of performance measures in forecast competitions.

Variable Length Memory Chains (VLMC), which are generalizations of finite order Markov chains, are an essential tool to modelize random sequences in many domains, as well as an interesting object in contemporary probability theory. The question of existence of stationary probability measures leads us to introduce a key combinatorial structure for words produced by a VLMC: the Longest Internal Suffix. This notion allows us to state a necessary and sufficient condition for a VLMC to admit a unique invariant probability measure.

This condition turns out to get a much simpler form for a subclass of VLMC: the stable VLMC. This natural subclass, unlike the general case, enjoys a renewal property. Namely, a stable VLMC induces a semi-Markov chain on an at most countable state space. Unfortunately, this discrete time renewal process does not contain the whole information of the VLMC, preventing the study of a stable VLMC to be reduced to the study of its induced semi-Markov chain. For a subclass of stable VLMC, the convergence in distribution of a VLMC towards its stationary probability measure is established.

Finally, finite state space semi-Markov chains turn out to be very special stable VLMC, shedding some new light on their limit distributions.

Constructions of numerous approximate sampling algorithms are based on the well-known fact that certain Gibbs measures are stationary distributions of ergodic stochastic differential equations (SDEs) driven by the Brownian motion. However, for some heavy-tailed distributions it can be shown that the associated SDE is not exponentially ergodic and that related sampling algorithms may perform poorly. A natural idea that has recently been explored in the machine learning literature in this context is to make use of stochastic processes with heavy tails instead of the Brownian motion. In this paper, we provide a rigorous theoretical framework for studying the problem of approximating heavy-tailed distributions via ergodic SDEs driven by symmetric (rotationally invariant) α-stable processes.

We consider the problem of delay estimation by the observations of the solutions of several SDEs. It is known that the MLEs for these models are consistent and asymptotically normal, but the likelihood ratio functions are not differentiable w.r.t. the parameter, and therefore the numerical calculation of the MLEs encounter certain difficulties. We propose One-step and Two-step MLEs, whose calculation has no such problems and provide an estimator asymptotically equivalent to the MLE. These constructions are realized in two or three steps. First, we construct preliminary estimators which are consistent and asymptotically normal, but not asymptotically efficient. Then we use these estimators and a modified Fisher-score device to obtain One-step and Two-step MLEs. We suppose that its numerical realization is much more simple. Stochastic Pantograph equation is introduced and related statistical problems are discussed.

Graphical models are a useful tool with increasing diffusion. In the categorical variable framework, they provide important visual support to understand the relationships among the considered variables. Besides, particular chain graphical models are suitable to represent multivariate regression models. However, the associated parameterization, such as marginal log-linear models, is often difficult to interpret when the number of variables increases because of a large number of parameters involved. On the contrary, conditional and marginal independencies reduce the number of parameters needed to represent the joint probability distribution of the variables. In compliance with the parsimonious principle, it is worthwhile to consider also the so-called context-specific independencies, which are conditional independencies holding for particular values of the variables in the conditioning set. In this work, we propose a particular chain graphical model able to represent these context-specific independencies through labeled arcs. We provide also the Markov properties able to describe marginal, conditional, and context-specific independencies from this new chain graph. Finally, we show the results in an application to a real data set.

A consequence of the celebrated Dvoretzky–Milman theorem is a strong law of large numbers for d-dimensional random projections of the n-dimensional cube. It shows that, with respect to the Hausdorff distance, a uniform random projection of the cube ${[-1/\sqrt{\mathit{n}},+1/\sqrt{\mathit{n}}]}^{\mathit{n}}$ onto ${\mathbb{R}}^{\mathit{d}}$ converges almost surely to a centered d-dimensional Euclidean ball of radius $\sqrt{2/\mathit{\pi }}$, as $\mathit{n}\to \infty$. We start by providing an alternative proof of this strong law via the Artstein–Vitale law of large numbers for random compact sets. Then, for every point inside the ball of radius $\sqrt{2/\mathit{\pi }}$, we determine the asymptotic number of vertices and the volume of the part of the cube projected ‘close’ to this point. More generally, we study large deviations for random projections of arbitrary product measures. Let ${\mathit{\nu }}^{\otimes \mathit{n}}$ be the n-fold product measure of a Borel probability measure ν on $\mathbb{R}$, and let I be uniformly distributed on the Stiefel manifold of orthogonal d-frames in ${\mathbb{R}}^{\mathit{n}}$. It is shown that the sequence of random measures ${\mathit{\nu }}^{\otimes \mathit{n}}\circ {({\mathit{n}}^{-1/2}{\mathit{I}}^{\ast })}^{-1}$, $\mathit{n}\in \mathbb{N}$, satisfies a large deviation principle with probability 1. The rate function is explicitly identified in terms of the moment generating function of ν. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube ${[-1,1]}^{\mathit{n}}$ and the discrete cube ${\{-1,1\}}^{\mathit{n}}$ as special cases.

We propose an estimator for the mean of random variables in separable real Banach spaces using the empirical characteristic function. Assuming that the covariance operator of the random variable is bounded in a precise sense, we show that the proposed estimator achieves the optimal sub-Gaussian rate, except for a faster decaying mean-dependent term. Under a mild condition, an iterative refinement procedure can essentially eliminate the mean-dependent term and provide a shift-equivariant estimate. Our results particularly suggests that a certain Gaussian width that appears in the best known rate in the literature might not be necessary. Furthermore, using the boundedness of the characteristic functions, we also show that, except possibly at high signal-to-noise ratios, the proposed estimator is gracefully robust against adversarial “contamination”. Our analysis is overall concise and transparent, thanks to the tractability of the characteristic functions.