We study the problem of sampling from a probability distribution $\pi $ on $\mathbb{R}^{d}$ which has a density w.r.t. the Lebesgue measure known up to a normalization factor $x\mapsto \mathrm{e}^{-U(x)}/\int _{\mathbb{R}^{d}}\mathrm{e}^{-U(y)}\,\mathrm{d}y$. We analyze a sampling method based on the Euler discretization of the Langevin stochastic differential equations under the assumptions that the potential $U$ is continuously differentiable, $\nabla U$ is Lipschitz, and $U$ is strongly concave. We focus on the case where the gradient of the log-density cannot be directly computed but unbiased estimates of the gradient from possibly dependent observations are available. This setting can be seen as a combination of a stochastic approximation (here stochastic gradient) type algorithms with discretized Langevin dynamics. We obtain an upper bound of the Wasserstein-2 distance between the law of the iterates of this algorithm and the target distribution $\pi $ with constants depending explicitly on the Lipschitz and strong convexity constants of the potential and the dimension of the space. Finally, under weaker assumptions on $U$ and its gradient but in the presence of independent observations, we obtain analogous results in Wasserstein-2 distance.

It is known that backward iterations of independent copies of a contractive random Lipschitz function converge almost surely under mild assumptions. By a sieving (or thinning) procedure based on adding to the functions time and space components, it is possible to construct a scale invariant stochastic process. We study its distribution and paths properties. In particular, we show that it is càdlàg and has finite total variation. We also provide examples and analyse various properties of particular sieved iterative function systems including perpetuities and infinite Bernoulli convolutions, iterations of maximum, and random continued fractions.

A simple construction of adaptive confidence sets is proposed in isotonic, convex and unimodal regression. In univariate isotonic regression, the proposed confidence set enjoys uniform coverage over all non-decreasing regression functions. Furthermore, the diameter of the proposed confidence set automatically adapts to the unknown number of pieces of the true parameter, in the sense that the diameter is bounded from above by the minimax risk over the class of $k$-piecewise constant functions. The diameter of the confidence set is a simple increasing function of the number of jumps of the isotonic least-squares estimate.

A similar construction is proposed in convex regression where the true regression function is convex and piecewise affine. Here, the confidence set enjoys uniform coverage and its diameter automatically adapts to the number of affine pieces of the true regression function. The diameter of the confidence set is an increasing function of the number of affine pieces of the convex least-squares estimate.

We explain how to extend this technique to a non-convex set by proposing a similar adaptive confidence set in unimodal regression. The confidence set automatically adapts to the number of jumps of the true unimodal regression function and its diameter is an increasing function of the number of jumps of the unimodal least-squares estimate.

In this paper, we investigate a parking process on a uniform random rooted plane tree with $n$ vertices. Every vertex of the tree has a parking space for a single car. Cars arrive at independent uniformly random vertices of the tree. If the parking space at a vertex is unoccupied when a car arrives there, it parks. If not, the car drives towards the root and parks in the first empty space it encounters (if there is one). We are interested in asymptotics of the probability of the event that all cars can park when $\lfloor\alpha n\rfloor$ cars arrive, for $\alpha>0$. We observe that there is a phase transition at $\alpha_{c}:=\sqrt{2}-1$: if $\alpha<\alpha_{c}$ then the event has positive limiting probability, whereas for $\alpha>\alpha_{c}$ its probability tends to 0. Analogous results have been proved by Lackner and Panholzer (J. Combin. Theory Ser. A 142 (2016) 1–28), Goldschmidt and Przykucki (Combin. Probab. Comput. 28 (2019) 23–45) and Jones (J. Appl. Probab. 56 (2019) 1065–1085) for different underlying random tree models.

In this paper, we study an inference problem in generalized fractional Ornstein–Uhlenbeck (O–U) processes with an unknown change-point when the drift parameter is suspected to satisfy some constraints. The constraint considered is very general and, the testing problem studied generalizes a very recent inference problem in generalized O–U processes. We derive the unrestricted estimator (UE) and the restricted estimator (RE) and we establish the asymptotic properties of the UE and RE. We also propose some shrinkage-type estimators (SEs) as well as a test for testing the constraint. Finally, we derive the asymptotic power of the proposed test and we study the relative risk dominance of the proposed estimators.

A discrete statistical model is a subset of a probability simplex. Its maximum likelihood estimator (MLE) is a retraction from that simplex onto the model. We characterize all models for which this retraction is a rational function. This is a contribution via real algebraic geometry which rests on results on Horn uniformization due to Huh and Kapranov. We present an algorithm for constructing models with rational MLE, and we demonstrate it on a range of instances. Our focus lies on models familiar to statisticians, like Bayesian networks, decomposable graphical models and staged trees.

Surface integrals on density level sets often appear in asymptotic results of nonparametric level set estimation, such as for confidence regions and bandwidth selection. Also surface integrals can be used to describe the shape of level sets. Assuming the integrands are known, we consider three estimators of the surface integrals on density level sets, one as a direct plug-in estimator, and the other two based on different neighborhoods of level sets. For all the three estimators, we derive the rates of convergence and asymptotic distributions.

Central limit theorems (CLTs) for high-dimensional random vectors with dimension possibly growing with the sample size have received a lot of attention in the recent times. Chernozhukov et al. (Ann. Probab. 45 (2017) 2309–2352) proved a Berry–Esseen type result for high-dimensional averages for the class of sparsely convex sets including hyperrectangles as a special case and they proved that the rate of convergence can be upper bounded by $n^{-1/6}$ up to a polynomial factor of $\log p$ (where $n$ represents the sample size and $p$ denotes the dimension). Convergence to zero of the bound requires $\log^{7}p=o(n)$. We improve upon their result, for hyperrectangles, which only requires $\log^{4}p=o(n)$ (in the best case). This improvement is made possible by a sharper dimension-free anti-concentration inequality for Gaussian process on a compact metric space. In addition, we prove two non-uniform variants of the high-dimensional CLT based on the large deviation and non-uniform CLT results for random variables in a Banach space by Bentkus, Rackauskas, and Paulauskas. We apply our results in the context of post-selection inference in linear regression and of empirical processes.

We correct an error in Theorem 2.1 in Bladt and Sørensen (Bernoulli 20 (2014) 645–675), where the initial distribution of an auxiliary diffusion process that is used to describe the distribution of the proposed approximate diffusion bridge is wrong. As a consequence, we also correct the pseudo marginal Metropolis-Hastings algorithm that has an exact diffusion bridge as its target distribution. The same auxiliary diffusion plays a central role in the algorithm.

Hawkes process is a class of simple point processes with self-exciting and clustering properties. Hawkes process has been widely applied in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study precise deviations for Hawkes processes for large time asymptotics, that strictly extends and improves the existing results in the literature. Numerical illustrations will also be provided.

In survival analysis, epidemiology and related fields there exists an increasing interest in statistical methods for doubly truncated data. Double truncation appears with interval sampling and other sampling schemes, and refers to situations in which the target variable is subject to two (left and right) random observation limits. Doubly truncated data require specific corrections for the observational bias, and this affects a variety of settings including the estimation of marginal and multivariate distributions, regression problems, and multi-state models. In this work multivariate Efron–Petrosian integrals for doubly truncated data are introduced. These integrals naturally arise when the goal is the estimation of the mean of a general transformation which involves the doubly truncated variable and covariates. An asymptotic representation of the Efron–Petrosian integrals as a sum of i.i.d. terms is derived and, from this, consistency and distributional convergence are established. As a by-product, uniform i.i.d. representations for the marginal nonparametric maximum likelihood estimator and its corresponding weighting process are provided. Applications to correlation analysis, regression, and competing risks models are presented. A simulation study is reported too.

We consider a more generalized spiked covariance matrix, which is a general non-negative definite matrix with the spiked eigenvalues scattered into spaces of a few bulks and the largest ones allowed to tend to infinity. The study is split into two cases by whether the maximum absolute value of the eigenvector of the corresponding spikes tends to zero or not. On one hand, if it is zero, a Generalized Four Moment Theorem (G4MT) is proposed by relaxing the matching of the 3rd and the 4th moment to the tail probability decaying with certain rate, which shows the universality of the asymptotic law for the spiked eigenvalues of the generalized spiked covariance model. On the other hand, if it is not zero, the matches of the third and fourth moments in usual four moment theorem are weakened to only requiring the match of the 4th moment. Moreover, by applying the results to the Central Limit Theorem (CLT) for the spiked eigenvalues of the generalized spiked covariance model, we successively remove the restrictive condition of block wise diagonal assumption on the population covariance matrix in the previous works. This condition implies an unrealistic fact that the spiked eigenvalues and bulked eigenvalues are generated by independent variables, respectively. Thus, the new CLT will have much better application domain.

We study the following equation \begin{equation*}\frac{\partial u(t,x)}{\partial t}=\Delta u(t,x)+b\bigl(u(t,x)\bigr)+\sigma \dot{W}(t,x),\quad t>0,\end{equation*} where $\sigma $ is a positive constant and $\dot{W}$ is a space–time white noise. The initial condition $u(0,x)=u_{0}(x)$ is assumed to be a nonnegative and continuous function. We first study the problem on $[0,1]$ with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of Bonder and Groisman in (Phys. D 238 (2009) 209–215), our first result shows that the solution blows up in finite time if and only if for some $a>0$, \begin{equation*}\int _{a}^{\infty }\frac{1}{b(s)}\,\mathrm{d}s<\infty,\end{equation*} which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in $\mathbf{R}^{d}$.

We consider the problem of designing robust numerical integration scheme of the solution of a one-dimensional SDE with non-globally Lipschitz drift and diffusion coefficients behaving as $x^{\alpha }$, with $\alpha >1$. We propose an (semi-explicit) exponential-Euler scheme for which we obtain a theoretical convergence rate for the weak error. To this aim, we analyze the $C^{1,4}$ regularity of the solution of the associated backward Kolmogorov PDE using its Feynman–Kac representation and the flow derivative of the involved processes. Under some suitable hypotheses on the parameters of the model, we prove a rate of weak convergence of order one for the proposed exponential Euler scheme, and illustrate it with some numerical experiments.

In this article, we discuss ergodicity properties of a diffusion process given through an Itô stochastic differential equation. We identify conditions on the drift and diffusion coefficients which result in sub-geometric ergodicity of the corresponding semigroup with respect to the total variation distance. We also prove sub-geometric contractivity and ergodicity of the semigroup under a class of Wasserstein distances. Finally, we discuss sub-geometric ergodicity of two classes of Markov processes with jumps.

Stationary subspace analysis (SSA) searches for linear combinations of the components of nonstationary vector time series that are stationary. These linear combinations and their number define an associated stationary subspace and its dimension. SSA is studied here for zero mean nonstationary covariance processes. We characterize stationary subspaces and their dimensions in terms of eigenvalues and eigenvectors of certain symmetric matrices. This characterization is then used to derive formal statistical tests for estimating dimensions of stationary subspaces. Eigenstructure-based techniques are also proposed to estimate stationary subspaces, without relying on previously used computationally intensive optimization-based methods. Finally, the introduced methodologies are examined on simulated and real data.

In this paper, we propose a test for the equality of multiple distributions based on kernel mean embeddings. Our framework provides a flexible way to handle multivariate data by virtue of kernel methods and allows the number of distributions to increase with the sample size. This is in contrast to previous studies that have been mostly restricted to classical univariate settings with a fixed number of distributions. By building on Cramér-type moderate deviation for degenerate two-sample $V$-statistics, we derive the limiting null distribution of the test statistic and show that it converges to a Gumbel distribution. The limiting distribution, however, depends on an infinite number of nuisance parameters, which makes it infeasible for use in practice. To address this issue, the proposed test is implemented via the permutation procedure and is shown to be minimax rate optimal against sparse alternatives. During our analysis, an exponential concentration inequality for the permuted test statistic is developed which may be of independent interest.

In this paper, we obtain uniform and non-uniform bounds on the Kolmogorov distance in the normal approximation for Jack deformations of the character ratio, by using Stein’s method and zero-bias couplings. Our uniform bound comes very close to that conjectured by Fulman (J. Combin. Theory Ser. A 108 (2004) 275–296). As a by-product of the proof of the non-uniform bound, we obtain a Rosenthal-type inequality for zero-bias couplings.

Due to the surge of data storage techniques, the need for the development of appropriate techniques to identify patterns and to extract knowledge from the resulting enormous data sets, which can be viewed as collections of dependent functional data, is of increasing interest in many scientific areas. We develop a similarity measure for spectral density operators of a collection of functional time series, which is based on the aggregation of Hilbert–Schmidt differences of the individual time-varying spectral density operators. Under fairly general conditions, the asymptotic properties of the corresponding estimator are derived and asymptotic normality is established. The introduced statistic lends itself naturally to quantify (dis)-similarity between functional time series, which we subsequently exploit in order to build a spectral clustering algorithm. Our algorithm is the first of its kind in the analysis of non-stationary (functional) time series and enables to discover particular patterns by grouping together ‘similar’ series into clusters, thereby reducing the complexity of the analysis considerably. The algorithm is simple to implement and computationally feasible. As a further application, we provide a simple test for the hypothesis that the second order properties of two non-stationary functional time series coincide.

We investigate relations between additive convolution semigroups and max-convolution semigroups through the law of large numbers for the free multiplicative convolution. Based on these relations, we give a formula related with the Belinschi–Nica semigroup and the max-Belinschi–Nica semigroup. Finally, we give several limit theorems for classical, free and Boolean extreme values.

We investigate the statistical complexity of estimating the parameters of a discrete-state Markov chain kernel from a single long sequence of state observations. In the finite case, we characterize (modulo logarithmic factors) the minimax sample complexity of estimation with respect to the operator infinity norm, while in the countably infinite case, we analyze the problem with respect to a natural entry-wise norm derived from total variation. We show that in both cases, the sample complexity is governed by the mixing properties of the unknown chain, for which, in the finite-state case, there are known finite-sample estimators with fully empirical confidence intervals.

We study the problem of change point localization for covariance matrices in high dimensions. We assume that we observe a sequence of independent and centered $p$-dimensional sub-Gaussian random vectors whose covariance matrices are piecewise constant, and only change at unknown times. We are concerned with the localization task of estimating the positions of the change points. In our analysis, we allow for all the model parameters to change with the sample size $n$, including the dimension $p$, the minimal spacing between consecutive change points $\Delta $, the maximal Orlicz-$\psi _{2}$ norm $B$ of the sample points and the magnitude $\kappa $ of the smallest distributional change, defined as the minimal operator norm of the difference between the covariance matrix at a change point and the covariance matrix at the previous time point.

We introduce two procedures, one based on the binary segmentation algorithm and the other on its popular extension known as wild binary segmentation, and demonstrate that, under suitable conditions, both procedures can consistently estimate the change points. In particular, our second algorithm, called Wild Binary Segmentation through Independent Projection (WBSIP), delivers a localization error of order $B^{4}\kappa ^{-2}\log (n)$, which is shown to be minimax rate optimal, save, possibly, for the $\log (n)$ term. WBSIP requires the model parameters to satisfy the scaling $\Delta \kappa ^{2}\gtrsim pB^{4}\log ^{1+\xi }(n)$, for any $\xi >0$, which we demonstrate to be essentially necessary, in the sense that no algorithm can guarantee consistent localization if $\Delta \kappa ^{2}\lesssim pB^{4}$. This result reveals an interesting phase transition effect separating parameter combinations for which the localization task is feasible from the ones for which it is not.

We derive sharp lower bounds for $\mathcal{L}^{p}$-functions on the $n$-dimensional unit hypercube in terms of their $p$-ths marginal moments. Such bounds are the unique solutions of a system of constrained nonlinear integral equations depending on the marginals. For square-integrable functions, the bounds have an explicit expression in terms of the second marginals moments.

We derive a dimension-free Hanson–Wright inequality for quadratic forms of independent sub-gaussian random variables in a separable Hilbert space. Our inequality is an infinite-dimensional generalization of the classical Hanson–Wright inequality for finite-dimensional Euclidean random vectors. We illustrate an application to the generalized $K$-means clustering problem for non-Euclidean data. Specifically, we establish the exponential rate of convergence for a semidefinite relaxation of the generalized $K$-means, which together with a simple rounding algorithm imply the exact recovery of the true clustering structure.

We consider one-dimensional biased voter models, where 1’s replace 0’s at a faster rate than the other way round, started in a Heaviside initial state describing the interface between two infinite populations of 0’s and 1’s. In the limit of weak bias, for a diffusively rescaled process, we consider a measure-valued process describing the local fraction of type 1 sites as a function of time. Under a finite second moment condition on the rates, we show that in the diffusive scaling limit there is a drifted Brownian path with the property that all but a vanishingly small fraction of the sites on the left (resp. right) of this path are of type 0 (resp. 1). This extends known results for unbiased voter models. Our proofs depend crucially on recent results about interface tightness for biased voter models.

Gaussian graphical models are a popular tool to learn the dependence structure in the form of a graph among variables of interest. Bayesian methods have gained in popularity in the last two decades due to their ability to simultaneously learn the covariance and the graph. There is a wide variety of model-based methods to learn the underlying graph assuming various forms of the graphical structure. Although for scalability of the Markov chain Monte Carlo algorithms, decomposability is commonly imposed on the graph space, its possible implication on the posterior distribution of the graph is not clear. An open problem in Bayesian decomposable structure learning is whether the posterior distribution is able to select a meaningful decomposable graph that is “close” to the true non-decomposable graph, when the dimension of the variables increases with the sample size. In this article, we explore specific conditions on the true precision matrix and the graph, which results in an affirmative answer to this question with a commonly used hyper-inverse Wishart prior on the covariance matrix and a suitable complexity prior on the graph space. In absence of structural sparsity assumptions, our strong selection consistency holds in a high-dimensional setting where $p=O(n^{\alpha })$ for $\alpha <1/3$. We show when the true graph is non-decomposable, the posterior distribution concentrates on a set of graphs that are minimal triangulations of the true graph.

This paper develops a new class of functional depths. A generic member of this class is coined $J$th order $k$th moment integrated depth. It is based on the distribution of the cross-sectional halfspace depth of a function in the marginal evaluations (in time) of the random process. Asymptotic properties of the proposed depths are provided: we show that they are uniformly consistent and satisfy an inequality related to the law of the iterated logarithm. Moreover, limiting distributions are derived under mild regularity assumptions. The versatility displayed by the new class of depths makes them particularly amenable for capturing important features of functional distributions. This is illustrated in supervised learning, where we show that the corresponding maximum depth classifiers outperform classical competitors.

In a Bayesian framework, to make predictions on a sequence $X_{1},X_{2},\ldots $ of random observations, the inferrer needs to assign the predictive distributions $\sigma _{n}(\cdot )=P(X_{n+1}\in \cdot \mid X_{1},\ldots,X_{n})$. In this paper, we propose to assign $\sigma _{n}$ directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability has to be assessed. The data sequence $(X_{n})$ is assumed to be conditionally identically distributed (c.i.d.) in the sense of (Ann. Probab. 32 (2004) 2029–2052). To realize this programme, a class $\Sigma $ of predictive distributions is introduced and investigated. Such a $\Sigma $ is rich enough to model various real situations and $(X_{n})$ is actually c.i.d. if $\sigma _{n}$ belongs to $\Sigma $. Furthermore, when a new observation $X_{n+1}$ becomes available, $\sigma _{n+1}$ can be obtained by a simple recursive update of $\sigma _{n}$. If $\mu $ is the a.s. weak limit of $\sigma _{n}$, conditions for $\mu $ to be a.s. discrete are provided as well.