In this paper, we develop non-stationary martingale techniques for dependent data. We shall stress the non-stationary version of the projective Maxwell–Woodroofe condition, which will be essential for obtaining maximal inequalities and functional central limit theorem for the following examples: nonstationary $\rho$-mixing sequences, functions of linear processes with non-stationary innovations, locally stationary processes, quenched version of the functional central limit theorem for a stationary sequence, evolutions in random media such as a process sampled by a shifted Markov chain.

The main objective of the paper is to study the long-time behavior of general discrete dynamics driven by an ergodic stationary Gaussian noise. In our main result, we prove existence and uniqueness of the invariant distribution and exhibit some upper-bounds on the rate of convergence to equilibrium in terms of the asymptotic behavior of the covariance function of the Gaussian noise (or equivalently to its moving average representation). Then, we apply our general results to fractional dynamics (including the Euler Scheme associated to fractional driven Stochastic Differential Equations). When the Hurst parameter $H$ belongs to $(0,1/2)$ we retrieve, with a slightly more explicit approach due to the discrete-time setting, the rate exhibited by Hairer in a continuous time setting (Ann. Probab. 33 (2005) 703–758). In this fractional setting, we also emphasize the significant dependence of the rate of convergence to equilibrium on the local behaviour of the covariance function of the Gaussian noise.

We study the monotone single index model where a real response variable $Y$ is linked to a $d$-dimensional covariate $X$ through the relationship $E[Y|X]=\Psi_{0}(\alpha^{T}_{0}X)$, almost surely. Both the ridge function, $\Psi_{0}$, and the index parameter, $\alpha_{0}$, are unknown and the ridge function is assumed to be monotone. Under some appropriate conditions, we show that the rate of convergence in the $L_{2}$-norm for the least squares estimator of the bundled function $\Psi_{0}({\alpha}^{T}_{0}\cdot)$ is $n^{1/3}$. A similar result is established for the isolated ridge function, and the index is shown to converge at least at the rate $n^{1/3}$. Since the least squares estimator of the index is computationally intensive, we also consider alternative estimators of the index $\alpha_{0}$ from earlier literature. Moreover, we show that if the rate of convergence of such an alternative estimator is at least $n^{1/3}$, then the corresponding least-squares type estimators (obtained via a “plug-in” approach) of both the bundled and isolated ridge functions still converge at the rate $n^{1/3}$.

We propose an adaptively weighted group Lasso procedure for simultaneous variable selection and structure identification for varying coefficient quantile regression models and additive quantile regression models with ultra-high dimensional covariates. Under a strong sparsity condition, we establish selection consistency of the proposed Lasso procedure when the weights therein satisfy a set of general conditions. This consistency result, however, is reliant on a suitable choice of the tuning parameter for the Lasso penalty, which can be hard to make in practice. To alleviate this difficulty, we suggest a BIC-type criterion, which we call high-dimensional information criterion (HDIC), and show that the proposed Lasso procedure with the tuning parameter determined by HDIC still achieves selection consistency. Our simulation studies support strongly our theoretical findings.

This work deals with systems of interacting reinforced stochastic processes, where each process $X^{j}=(X_{n,j})_{n}$ is located at a vertex $j$ of a finite weighted direct graph, and it can be interpreted as the sequence of “actions” adopted by an agent $j$ of the network. The interaction among the evolving dynamics of these processes depends on the weighted adjacency matrix $W$ associated to the underlying graph: indeed, the probability that an agent $j$ chooses a certain action depends on its personal “inclination” $Z_{n,j}$ and on the inclinations $Z_{n,h}$, with $h\neq j$, of the other agents according to the elements of $W$.

Asymptotic results for the stochastic processes of the personal inclinations $Z^{j}=(Z_{n,j})_{n}$ have been subject of studies in recent papers (e.g., Aletti, Crimaldi and Ghiglietti [Ann. Appl. Probab. 27 (2017) 3787–3844], Crimaldi et al. [Synchronization and functional central limit theorems for interacting reinforced random walks (2019)]); while the asymptotic behavior of quantities based on the stochastic processes $X^{j}$ of the actions has never been studied yet. In this paper, we fill this gap by characterizing the asymptotic behavior of the empirical means $N_{n,j}=\sum_{k=1}^{n}X_{k,j}/n$, proving their almost sure synchronization and some central limit theorems in the sense of stable convergence. Moreover, we discuss some statistical applications of these convergence results concerning confidence intervals for the random limit toward which all the processes of the system almost surely converge and tools to make inference on the matrix $W$.

Most theoretical results on the relative errors of saddlepoint approximations in the extreme tails have involved placing conditions on the density/mass function. Checking the validity of such conditions is problematic when density/mass functions are intractable, as is typically the case in important practical applications involving convolved, compound, and first-passage distributions as well as for moment generating functions MGFs that are regularly varying. In this paper, we present novel conditions which ensure the existence of positive finite limiting relative errors for saddlepoint density/mass function and survival function approximations. These conditions, which are rather weak, are expressed entirely in terms of the MGF, hence the description purely Tauberian. We focus mainly on the cases in which there are positive and negative gamma distributional limits (the only other non-degenerate possibility being a Gaussian limit) and we show how to check the new conditions in important classes of models in these two settings.

We study a nonparametric likelihood ratio test (NPLRT) for Gaussian mixtures. It is based on the nonparametric maximum likelihood estimator in the context of demixing. The test concerns if a random sample is from the standard normal distribution. We consider mixing distributions of unbounded support for alternative hypothesis. We prove that the divergence rate of the NPLRT under the null is bounded by $\log n$, provided that the support range of the mixing distribution increases no faster than $(\log n/\log 9)^{1/2}$. We prove that the rate of $\sqrt{\log n}$ is a lower bound for the divergence rate if the support range increases no slower than the order of $\sqrt{\log n}$. Implications of the upper bound for the rate of divergence are discussed.

We obtain a Bernstein-type inequality for sums of Banach-valued random variables satisfying a weak dependence assumption of general type and under certain smoothness assumptions of the underlying Banach norm. We use this inequality in order to investigate in the asymptotical regime the error upper bounds for the broad family of spectral regularization methods for reproducing kernel decision rules, when trained on a sample coming from a $\tau$-mixing process.

We consider the problem of estimating a vector ${\boldsymbol{\mu}}=(\mu_{1},\dots,\mu_{n})$ under a squared loss, based on independent observations $Y_{i}\sim N(\mu_{i},1)$, $i=1,\dots,n$, and possibly extra structural assumptions. We argue that many estimators are asymptotically equal to $\hat{\mu}_{i}=\alpha\tilde{\mu}_{i}+(1-\alpha)Y_{i}+\xi_{i}=\tilde{\mu}_{i}+(1-\alpha)(Y_{i}-\tilde{\mu}_{i})+\xi_{i}$, where $\alpha\in[0,1]$ and $\tilde{\mu}_{i}$ may depend on the data, but is not a function of $Y_{i}$, and $\sum\xi_{i}^{2}=o_{p}(n)$.

We consider the optimal estimator of the form $\tilde{\mu}_{i}+g(Y_{i}-\tilde{\mu}_{i})$ for a general, possibly random, function $g$, and approximate it using nonparametric empirical Bayes ideas and techniques. We consider both the retrospective and the sequential estimation problems. We elaborate and demonstrate our results on the case where $\hat{\mu}_{i}$ are Kalman filter estimators. Simulations and a real data analysis are also provided.

In this article, we prove that for any probability distribution $\mu $ on $\mathbb{N}$ one can construct a two-sided stationary version of the infinite-bin model – an interacting particle system introduced by Foss and Konstantopoulos – with move distribution $\mu $. Using this result, we obtain a new formula for the speed of the front of infinite-bin models, as a series of positive terms. This implies that the growth rate $C(p)$ of the longest path in a Barak–Erdős graph of parameter $p$ is analytic on $(0,1]$.

We consider infinite-dimensional Hilbert space-valued random variables that are assumed to be temporal dependent in a broad sense. We prove a central limit theorem for the moving block bootstrap and for the tapered block bootstrap, and show that these block bootstrap procedures also provide consistent estimators of the long run covariance operator. Furthermore, we consider block bootstrap-based procedures for fully functional testing of the equality of mean functions between several independent functional time series. We establish validity of the block bootstrap methods in approximating the distribution of the statistic of interest under the null and show consistency of the block bootstrap-based tests under the alternative. The finite sample behaviour of the procedures is investigated by means of simulations. An application to a real-life dataset is also discussed.

This paper is devoted to establishing exponential bounds for the probabilities of deviation of a sample sum from its expectation, when the variables involved in the summation are obtained by sampling in a finite population according to a rejective scheme, generalizing simple random sampling without replacement, and by using an appropriate normalization. In contrast to Poisson sampling, classical deviation inequalities in the i.i.d. setting do not straightforwardly apply to sample sums related to rejective schemes, due to the inherent dependence structure of the sampled points. We show here how to overcome this difficulty, by combining the formulation of rejective sampling as Poisson sampling conditioned upon the sample size with the Esscher transformation. In particular, the Bennett/Bernstein type bounds thus established highlight the effect of the asymptotic variance of the (properly standardized) sample weighted sum and are shown to be much more accurate than those based on the negative association property shared by the terms involved in the summation. Beyond its interest in itself, such a result for rejective sampling is crucial, insofar as it permit to obtain tail bounds for many other sampling schemes, namely those that can be accurately approximated by rejective plans in the sense of the total variation distance.

Determinantal Point Processes (DPPs) are popular models for point processes with repulsion. They appear in numerous contexts, from physics to graph theory, and display appealing theoretical properties. On the more practical side of things, since DPPs tend to select sets of points that are some distance apart (repulsion), they have been advocated as a way of producing random subsets with high diversity. DPPs come in two variants: fixed-size and varying-size. A sample from a varying-size DPP is a subset of random cardinality, while in fixed-size “$k$-DPPs” the cardinality is fixed. The latter makes more sense in many applications, but unfortunately their computational properties are less attractive, since, among other things, inclusion probabilities are harder to compute. In this work, we show that as the size of the ground set grows, $k$-DPPs and DPPs become equivalent, in the sense that fixed-order inclusion probabilities converge. As a by-product, we obtain saddlepoint formulas for inclusion probabilities in $k$-DPPs. These turn out to be extremely accurate, and suffer less from numerical difficulties than exact methods do. Our results also suggest that $k$-DPPs and DPPs also have equivalent maximum likelihood estimators. Finally, we obtain results on asymptotic approximations of elementary symmetric polynomials which may be of independent interest.

We consider a $p$-dimensional time series where the dimension $p$ increases with the sample size $n$. The resulting data matrix $\mathbf{X}$ follows a stochastic volatility model: each entry consists of a positive random volatility term multiplied by an independent noise term. The volatility multipliers introduce dependence in each row and across the rows. We study the asymptotic behavior of the eigenvalues and eigenvectors of the sample covariance matrix $\mathbf{X}\mathbf{X}'$ under a regular variation assumption on the noise. In particular, we prove Poisson convergence for the point process of the centered and normalized eigenvalues and derive limit theory for functionals acting on them, such as the trace. We prove related results for stochastic volatility models with additional linear dependence structure and for stochastic volatility models where the time-varying volatility terms are extinguished with high probability when $n$ increases. We provide explicit approximations of the eigenvectors which are of a strikingly simple structure. The main tools for proving these results are large deviation theorems for heavy-tailed time series, advocating a unified approach to the study of the eigenstructure of heavy-tailed random matrices.

This article presents a limit theorem for the gaps $\widehat{G}_{i:n}:=X_{n-i+1:n}-X_{n-i:n}$ between order statistics $X_{1:n}\le\cdots\le X_{n:n}$ of a sample of size $n$ from a random discrete distribution on the positive integers $(P_{1},P_{2},\ldots)$ governed by a residual allocation model (also called a Bernoulli sieve) $P_{j}:=H_{j}\prod_{i=1}^{j-1}(1-H_{i})$ for a sequence of independent random hazard variables $H_{i}$ which are identically distributed according to some distribution of $H\in(0,1)$ such that $-\log(1-H)$ has a non-lattice distribution with finite mean $\mu_{\log}$. As $n\to\infty$ the finite dimensional distributions of the gaps $\widehat{G}_{i:n}$ converge to those of limiting gaps $G_{i}$ which are the numbers of points in a stationary renewal process with i.i.d. spacings $-\log(1-H_{j})$ between times $T_{i-1}$ and $T_{i}$ of births in a Yule process, that is $T_{i}:=\sum_{k=1}^{i}\varepsilon_{k}/k$ for a sequence of i.i.d. exponential variables $\varepsilon_{k}$ with mean 1. A consequence is that the mean of $\widehat{G}_{i:n}$ converges to the mean of $G_{i}$, which is $1/(i\mu_{\log})$. This limit theorem simplifies and extends a result of Gnedin, Iksanov and Roesler for the Bernoulli sieve.

We consider random walks $\lambda $-biased towards the root on a Galton–Watson tree, whose offspring distribution $(p_{k})_{k\geq 1}$ is non-degenerate and has finite mean $m>1$. In the transient regime $0<\lambda <m$, the loop-erased trajectory of the biased random walk defines the $\lambda $-harmonic ray, whose law is the $\lambda $-harmonic measure on the boundary of the Galton–Watson tree. We answer a question of Lyons, Pemantle and Peres (In Classical and Modern Branching Processes (Minneapolis, MN, 1994) (1997) 223–237 Springer) by showing that the $\lambda $-harmonic measure has a.s. strictly larger Hausdorff dimension than the visibility measure, which is the harmonic measure corresponding to the simple forward random walk. We also prove that the average number of children of the vertices along the $\lambda $-harmonic ray is a.s. bounded below by $m$ and bounded above by $m^{-1}\sum k^{2}p_{k}$. Moreover, at least for $0<\lambda \leq 1$, the average number of children of the vertices along the $\lambda $-harmonic ray is a.s. strictly larger than that of the $\lambda $-biased random walk trajectory. We observe that the latter is not monotone in the bias parameter $\lambda $.

For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polynomials. To that purpose, we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.

We consider the Boolean model on $\mathbb{R}^{d}$. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let $C$ denote the connected component of the origin in the Boolean model. Let $|C|$ denotes its volume. Let $\ell$ denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that $\mathbb{E}(|C|)$ is finite if and only if there exists $A,B>0$ such that $\mathbb{P}(\ell\ge n)\le Ae^{-Bn}$ for all $n\ge1$.

This paper aims at semi-parametrically estimating the input process to a Lévy-driven queue by sampling the workload process at Poisson times. We construct a method-of-moments based estimator for the Lévy process’ characteristic exponent. This method exploits the known distribution of the workload sampled at an exponential time, thus taking into account the dependence between subsequent samples. Verifiable conditions for consistency and asymptotic normality are provided, along with explicit expressions for the asymptotic variance. The method requires an intermediate estimation step of estimating a constant (related to both the input distribution and the sampling rate); this constant also features in the asymptotic analysis. For subordinator Lévy input, a partial MLE is constructed for the intermediate step and we show that it satisfies the consistency and asymptotic normality conditions. For general spectrally-positive Lévy input a biased estimator is proposed that only uses workload observations above some threshold; the bias can be made arbitrarily small by appropriately choosing the threshold.

Consider the following nonparametric transformation model $\Lambda (Y)=m(X)+\varepsilon $, where $X$ is a $d$-dimensional covariate, $Y$ is a continuous univariate dependent variable and $\varepsilon $ is an error term with zero mean and which is independent of $X$. We assume that the unknown transformation $\Lambda $ is strictly increasing and that $m$ is an unknown regression function. Our goal is to develop two new nonparametric estimators of the transformation $\Lambda $, the first one based on the least squares loss and the second one based on the least absolute deviation loss, and to compare their performance with that of the estimators developed by Chiappori, Komunjer and Kristensen (J. Econometrics 188 (2015) 22–39). Our proposed estimators are based on an estimator of the conditional distribution of $U$ given $X$, where $U$ is an appropriate transformation of $Y$ that is uniformly distributed. The main motivation for working with $U$ instead of $Y$ is that, in transformation models, the response $Y$ is often skewed with very long tails, and so kernel smoothing based on $Y$ does not work well. Hence, we expect to obtain better estimators if we pre-transform $Y$ before applying kernel smoothing. We establish the asymptotic normality of the two proposed estimators. We also carry out a simulation study to illustrate the performance of our estimators, to compare these new estimators with the ones of Chiappori, Komunjer and Kristensen (J. Econometrics 188 (2015) 22–39) and to see under which model conditions which estimators behave the best.

Let $X$ be a symmetric jump process on $\mathbb{R}^{d}$ such that the corresponding jumping kernel $J(x,y)$ satisfies

\[J(x,y)\le\frac{c}{|x-y|^{d+2}\log^{1+\varepsilon}(e+|x-y|)}\] for all $x,y\in\mathbb{R}^{d}$ with $|x-y|\ge1$ and some constants $c,\varepsilon>0$. Under additional mild assumptions on $J(x,y)$ for $|x-y|<1$, we show that $C\sqrt{r\log\log r}$ with some constant $C>0$ is an upper rate function of the process $X$, which enjoys the same form as that for Brownian motions. The approach is based on heat kernel estimates of large time for the process $X$. As a by-product, we also obtain two-sided heat kernel estimates of large time for symmetric jump processes whose jumping kernels are comparable to

\[\frac{1}{|x-y|^{d+2+\varepsilon}}\] for all $x,y\in\mathbb{R}^{d}$ with $|x-y|\ge1$ and some constant $\varepsilon>0$.

Markov chain Monte Carlo is widely used in a variety of scientific applications to generate approximate samples from intractable distributions. A thorough understanding of the convergence and mixing properties of these Markov chains can be obtained by studying the spectrum of the associated Markov operator. While several methods to bound/estimate the second largest eigenvalue are available in the literature, very few general techniques for consistent estimation of the entire spectrum have been proposed. Existing methods for this purpose require the Markov transition density to be available in closed form, which is often not true in practice, especially in modern statistical applications. In this paper, we propose a novel method to consistently estimate the entire spectrum of a general class of Markov chains arising from a popular and widely used statistical approach known as Data Augmentation. The transition densities of these Markov chains can often only be expressed as intractable integrals. We illustrate the applicability of our method using real and simulated data.

The paper is concerned with asymptotic properties of the principal components analysis of functional data. The currently available results assume the existence of the fourth moment. We develop analogous results in a setting which does not require this assumption. Instead, we assume that the observed functions are regularly varying. We derive the asymptotic distribution of the sample covariance operator and of the sample functional principal components. We obtain a number of results on the convergence of moments and almost sure convergence. We apply the new theory to establish the consistency of the regression operator in a functional linear model.

We study the problem of matrix estimation and matrix completion under a general framework. This framework includes several important models as special cases such as the Gaussian mixture model, mixed membership model, bi-clustering model and dictionary learning. We establish the optimal convergence rates in a minimax sense for estimation of the signal matrix under the Frobenius norm and under the spectral norm. As a consequence of our general result we obtain minimax optimal rates of convergence for various special models.

The concept of Rademacher complexity for independent sequences of random variables is extended to Markov chains. The proposed notion of “regenerative block Rademacher complexity” (of a class of functions) follows from renewal theory and allows to control the expected values of suprema (over the class of functions) of empirical processes based on Harris Markov chains as well as the excess probability. For classes of Vapnik–Chervonenkis type, bounds on the “regenerative block Rademacher complexity” are established. These bounds depend essentially on the sample size and the probability tails of the regeneration times. The proposed approach is employed to obtain convergence rates for the kernel density estimator of the stationary measure and to derive concentration inequalities for the Metropolis–Hastings algorithm.

We study solutions to the stochastic fixed point equation $X\stackrel{d}{=}AX+B$ when the coefficients are nonnegative and $B$ is an “inverse exponential decay” ($\operatorname{IED}$) random variable. We provide theorems on the left tail of $X$ which complement well-known tail results of Kesten and Goldie. We generalize our results to ARMA processes with nonnegative coefficients whose noise terms are from the $\operatorname{IED}$ class. We describe the lower envelope for these ARMA processes.

We investigate links between the so-called Stein’s density approach in dimension one and some functional and concentration inequalities. We show that measures having a finite first moment and a density with connected support satisfy a weighted Poincaré inequality with the weight being the Stein kernel, that indeed exists and is unique in this case. Furthermore, we prove weighted log-Sobolev and asymmetric Brascamp–Lieb type inequalities related to Stein kernels. We also show that existence of a uniformly bounded Stein kernel is sufficient to ensure a positive Cheeger isoperimetric constant. Then we derive new concentration inequalities. In particular, we prove generalized Mills’ type inequalities when a Stein kernel is uniformly bounded and sub-gamma concentration for Lipschitz functions of a variable with a sub-linear Stein kernel. More generally, when some exponential moments are finite, the Laplace transform of the random variable of interest is shown to bounded from above by the Laplace transform of the Stein kernel. Along the way, we prove a general lemma for bounding the Laplace transform of a random variable, that may be of independent interest. We also provide density and tail formulas as well as tail bounds, generalizing previous results that where obtained in the context of Malliavin calculus.