Nonlinear regression models are a standard tool for modelling real phenomena, with several applications in machine learning, ecology, econometrics, and so forth. Estimating the parameters of these models has attracted a lot of attention during many years. We focus here on a recursive method for estimating parameters of nonlinear regressions. Indeed, these kinds of methods, whose most famous instances are probably the stochastic gradient algorithm and its averaged version, enable to deal efficiently with massive data arriving sequentially. Nevertheless, they can be, in practice, non-robust when the eigenvalues of the Hessian of the functional we would like to minimize are at different scales. To avoid this problem, we first focus on an online Stochastic Gauss-Newton algorithm. In order to improve the estimates behaviour in the case of bad initialization, we then introduce a new Averaged Stochastic Gauss-Newton algorithm and prove its asymptotic efficiency.

We study point processes that consist of certain centres of point tuples of an underlying Poisson process. Such processes arise in stochastic geometry in the study of exceedances of various functionals describing geometric properties of the Poisson process. We use a coupling of the point process with its Palm version to prove a general Poisson limit theorem. We then combine our general result with the theory of asymptotic shapes of large cells (Kendall’s problem) in random mosaics and prove Poisson limit theorems for large cells (with respect to a general size functional) in the Poisson-Voronoi and Poisson-Delaunay mosaic. As a consequence, we establish scaling limits for maxima of concrete size functionals. This extends extreme value results from (Extremes 17 (2014) 359–385) and (Stochastic Process. Appl. 124 (2014) 2917–2953).

We derive a Central Limit Theorem (CLT) for $log\left|\det \left(W_N-E_N\right)\right|$, where $W_N$ is a Wigner matrix, and $E_N$ is local to the edge of the semi-circle law. Precisely, $E_N=2+N^{-2∕3}{\mathit{\sigma }}_N$ with ${\mathit{\sigma }}_N$ being either a constant (possibly negative), or a sequence of positive real numbers, slowly diverging to infinity so that ${\mathit{\sigma }}_N\ll {log}^{2}N$. We also extend our CLT to cover spiked Wigner matrices. Our interest in the CLT is motivated by its applications to statistical testing in critically spiked models and to the fluctuations of the free energy in the spherical Sherrington-Kirkpatrick model of statistical physics.

We consider locally isotropic Gaussian random fields on the N-dimensional Euclidean space for fixed N. Using the so called Gaussian Orthogonally Invariant matrices first studied by Mallows in 1961 which include the celebrated Gaussian Orthogonal Ensemble (GOE), we establish the Kac–Rice representation of expected number of critical points of non-isotropic Gaussian fields, complementing the isotropic case obtained by Cheng and Schwartzman in 2018. In the limit $N=\mathrm{\infty }$, we show that such a representation can be always given by GOE matrices, as conjectured by Auffinger and Zeng in 2020.

Due to the lack of a canonical ordering in ${\mathbb{R}}^d$ for $d>1$, defining multivariate generalizations of the classical univariate ranks has been a long-standing open problem in statistics. Optimal transport has been shown to offer a solution in which multivariate ranks are obtained by transporting data points to a grid that approximates a uniform reference measure (Ann. Statist. 45 (2017) 223–256; Hallin (2017); Ann. Statist. 49 (2021) 1139–1165), thereby inducing ranks, signs, and a data-driven ordering of ${\mathbb{R}}^d$. We take up this new perspective to define and study multivariate analogues of the sign covariance/quadrant statistic, Spearman’s rho, Kendall’s tau, and van der Waerden covariances. The resulting tests of multivariate independence are fully distribution-free, hence uniformly valid irrespective of the actual (absolutely continuous) distribution of the observations. Our results provide the asymptotic distribution theory for these new test statistics, with asymptotic approximations to critical values to be used for testing independence between random vectors, as well as a power analysis of the resulting tests in an extension of the so-called (bivariate) Konijn model. This power analysis includes a multivariate Chernoff–Savage property guaranteeing that, under elliptical generalized Konijn models, the asymptotic relative efficiency of our van der Waerden tests with respect to Wilks’ classical (pseudo-)Gaussian procedure is strictly larger than or equal to one, where equality is achieved under Gaussian distributions only. We similarly provide a lower bound for the asymptotic relative efficiency of our Spearman procedure with respect to Wilks’ test, thus extending the classical result by Hodges and Lehmann on the asymptotic relative efficiency, in univariate location models, of Wilcoxon tests with respect to the Student ones.

Cramér’s moderate deviations give a quantitative estimate for the relative error of the normal approximation and provide theoretical justifications for many estimators used in statistics. In this paper, we establish self-normalized Cramér type moderate deviations for martingales under some mild conditions. The result extends an earlier work of Fan et al. (Bernoulli 25 (2019) 2793–2823). Moreover, applications of our result to Student’s statistic, stationary martingale difference sequences and branching processes in a random environment are also discussed.

We consider a non-ergodic class of stationary real harmonizable symmetric α-stable processes $X=\left\{X(t):t\in \mathbb{R}\right\}$ with a finite symmetric and absolutely continuous control measure. We refer to its density function as the spectral density of X. These processes admit a LePage series representation and are conditionally Gaussian, which allows us to derive the non-ergodic limit of sample functions on X. In particular, we give an explicit expression for the non-ergodic limits of the empirical characteristic function of X and the lag process $\left\{X(t+h)-X(t):t\in \mathbb{R}\right\}$ with $h>0$, respectively. The process admits an equivalent representation as a series of sinusoidal waves with random frequencies which are i.i.d. with the (normalized) spectral density of X as their probability density function. Based on strongly consistent frequency estimation using the periodogram we present a strongly consistent estimator of the spectral density. The computation of the periodogram is fast and efficient, and our method is not affected by the non-ergodicity of X.

In this paper we observe a set, possibly a continuum, of signals corrupted by noise. Each signal is a finite mixture of an unknown number of features belonging to a continuous dictionary. The continuous dictionary is parametrized by a real non-linear parameter. We shall assume that the signals share an underlying structure by assuming that each signal has its active features included in a finite and sparse set. We formulate regularized optimization problem to estimate simultaneously the linear coefficients in the mixtures and the non-linear parameters of the features. The optimization problem is composed of a data fidelity term and a $({\ell }_1,L^p)$-penalty. We call its solution the Group-Nonlinear-Lasso and provide high probability bounds on the prediction error using certificate functions. Following recent works on the geometry of off-the-grid methods, we show that such functions can be constructed provided the parameters of the active features are pairwise separated by a constant with respect to a Riemannian metric. When the number of signals is finite and the noise is assumed Gaussian, we give refinements of our results for $p=1$ and $p=2$ using tail bounds on suprema of Gaussian and ${\mathit{\chi }}^2$ random processes. When $p=2$, our prediction error reaches the rates obtained by the Group-Lasso estimator in the multi-task linear regression model. Furthermore, for $p=2$ these prediction rates are faster than for $p=1$ when all signals share most of the non-linear parameters.

We study stability and sample complexity properties of divergence regularized optimal transport (DOT). First, we obtain quantitative stability results for optimizers of DOT measured in Wasserstein distance, which are applicable to a wide class of divergences and simultaneously improve known results for entropic optimal transport. Second, we study the case of sample complexity, where the DOT problem is approximated using empirical measures of the marginals. We show that divergence regularization can improve the corresponding convergence rate compared to unregularized optimal transport. To this end, we prove upper bounds which exploit both the regularity of cost function and divergence functional, as well as the intrinsic dimension of the marginals. Along the way, we establish regularity properties of dual optimizers of DOT, as well as general limit theorems for empirical measures with suitable classes of test functions.

In this paper, we develop a general theory for adaptive nonparametric estimation of the mean function of a non-stationary and nonlinear time series model using deep neural networks (DNNs). We first consider two types of DNN estimators, non-penalized and sparse-penalized DNN estimators, and establish their generalization error bounds for general non-stationary time series. We then derive minimax lower bounds for estimating mean functions belonging to a wide class of nonlinear autoregressive (AR) models that include nonlinear generalized additive AR, single index, and threshold AR models. Building upon the results, we show that the sparse-penalized DNN estimator is adaptive and attains the minimax optimal rates up to a poly-logarithmic factor for many nonlinear AR models. Through numerical simulations, we demonstrate the usefulness of the DNN methods for estimating nonlinear AR models with intrinsic low-dimensional structures and discontinuous or rough mean functions, which is consistent with our theory.

The problem of simultaneously testing the marginal distributions of sequentially monitored, independent data streams is considered. The decisions for the various testing problems can be made at different times, using data from all streams, which can be monitored until all decisions have been made. Moreover, arbitrary a priori bounds are assumed on the number of signals, i.e., data streams in which the alternative hypothesis is correct. A novel sequential multiple testing procedure is proposed and it is shown to achieve the minimum expected decision time, simultaneously in every data stream and under every signal configuration, asymptotically as certain metrics of global error rates go to zero. This optimality property is established under general parametric composite hypotheses, various error metrics, and weak distributional assumptions that allow for temporal dependence. Furthermore, the limit of the factor by which the expected decision time in a data stream increases when one is limited to synchronous or decentralized procedures is evaluated. Finally, two existing sequential multiple testing procedures in the literature are compared with the proposed one in various simulation studies.

We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $\mathit{\alpha },\mathit{\beta }\ge 0$, $\mathit{\alpha }\ne \mathit{\beta }$,

$${\mathit{\psi }}_{\mathit{\alpha },\mathit{\beta }}(x,y)=\left(y\frac{1+\mathit{\beta }(x+y)}{1+\mathit{\alpha }x+\mathit{\beta }y},x\frac{1+\mathit{\alpha }(x+y)}{1+\mathit{\alpha }x+\mathit{\beta }y}\right),$$

then the random variables U and V defined by $(U,V)={\mathit{\psi }}_{\mathit{\alpha },\mathit{\beta }}(X,Y)$ are independent if and only if X and Y follow Kummer distributions with suitably related parameters. In other words, any invariant measure for a lattice recursion model governed by ${\mathit{\psi }}_{\mathit{\alpha },\mathit{\beta }}$ in the scheme introduced by Croydon and Sasada in (Croydon and Sasada (2020)) is necessarily a product measure with Kummer marginals. The result extends earlier characterizations of Kummer and gamma laws by independence of

$$U=\frac{Y}{1+X}\text{and}V=X\left(1+\frac{Y}{1+X}\right),$$

which corresponds to the case of ${\mathit{\psi }}_{1,0}$. We also show, in the supplement, that this independence property of Kummer laws covers, as limiting cases, several independence models known in the literature: the Lukacs, the Kummer-Gamma, the Matsumoto-Yor and the discrete Korteweg de Vries models.

In the regression framework, the empirical measure based on the responses resulting from the nearest neighbors, among the covariates, to a given point x is introduced and studied as a central statistical quantity. First, the associated empirical process is shown to satisfy a uniform central limit theorem under a local bracketing entropy condition on the underlying class of functions reflecting the localizing nature of the nearest neighbor algorithm. Second a uniform non-asymptotic bound is established under a well-known condition, often referred to as Vapnik-Chervonenkis, on the uniform entropy numbers. The covariance of the Gaussian limit obtained in the uniform central limit theorem is simply equal to the conditional covariance operator given the covariate value. This suggests the possibility of using standard formulas to estimate the variance by using only the nearest neighbors instead of the full data. This is illustrated on two problems: the estimation of the conditional cumulative distribution function and local linear regression.

This work aims at making a comprehensive contribution in the general area of parametric inference for discretely observed diffusion processes. Established approaches for likelihood-based estimation invoke a time-discretisation scheme for the approximation of the intractable transition dynamics of the Stochastic Differential Equation (SDE) model over finite time periods. The scheme is applied for a step-size $\mathit{\delta }>0$, that is either user-selected or determined by the data. Recent research has highlighted the critical effect of the choice of numerical scheme on the behaviour of derived parameter estimates in the setting of hypo-elliptic SDEs. In brief, in our work, first, we develop two weak second order sampling schemes (to cover both hypo-elliptic and elliptic SDEs) and produce a small time expansion for the density of the schemes to form a proxy for the true intractable SDE transition density. Then, we establish a collection of analytic results for likelihood-based parameter estimates obtained via the formed proxies, thus providing a theoretical framework that showcases advantages from the use of the developed methodology for SDE calibration. We present numerical results from carrying out classical or Bayesian inference, for both elliptic and hypo-elliptic SDEs.

We introduce a hull operator on Poisson point processes, the easiest example being the convex hull of the support of a point process in Euclidean space. Assuming that the intensity measure of the process is known on the set generated by the hull operator, we discuss estimation of an expected linear statistic built on the Poisson process. In special cases, our general scheme yields an estimator of the volume of a convex body or an estimator of an integral of a Hölder function. We show that the estimation error is given by the Kabanov–Skorohod integral with respect to the underlying Poisson process. A crucial ingredient of our approach is a spatial strong Markov property of the underlying Poisson process with respect to the hull. We derive the rate of normal convergence for the estimation error, and illustrate it on an application to estimators of integrals of a Hölder function. We also discuss estimation of higher order symmetric statistics.

Cross-validation is the standard approach for tuning parameter selection in many non-parametric regression problems. However its use is less common in change-point regression, perhaps as its prediction error-based criterion may appear to permit small spurious changes and hence be less well-suited to estimation of the number and location of change-points. We show that in fact the problems of cross-validation with squared error loss are more severe and can lead to systematic under- or over-estimation of the number of change-points, and highly suboptimal estimation of the mean function in simple settings where changes are easily detectable. We propose two simple approaches to remedy these issues, the first involving the use of absolute error rather than squared error loss, and the second involving modifying the holdout sets used. For the latter, we provide conditions that permit consistent estimation of the number of change-points for a general change-point estimation procedure. We show these conditions are satisfied for least squares estimation using new results on its performance when supplied with the incorrect number of change-points. Numerical experiments show that our new approaches are competitive with common change-point methods using classical tuning parameter choices when error distributions are well-specified, but can substantially outperform these in misspecified models. An implementation of our methodology is available in the $\mathtt{R}$ package $\mathtt{crossvalidationCP}$ on CRAN.

In 1926, G. Udny Yule (J. R. Stat. Soc. 89 (1926) 1–63) considered the following problem: given a sequence of pairs of random variables $\{X_k,Y_k\}$ ($k=1,2,\dots ,n$), and letting $X_i=S_i$ and $Y_i=S_i^{\prime }$ where $S_i$ and $S_i^{\prime }$ are the partial sums of two independent random walks, what is the distribution of the empirical correlation coefficient

$${\mathit{\rho }}_n=\frac{{\sum }_{i=1}^n{S}_i{S}_i^{\prime }-\frac{1}{n}({\sum }_{i=1}^n{S}_i)({\sum }_{i=1}^n{S}_i^{\prime })}{\sqrt{{\sum }_{i=1}^n{S}_i^2-\frac{1}{n}{({\sum }_{i=1}^n{S}_i)}^2}\sqrt{{\sum }_{i=1}^n{(S_i^{\prime })}^2-\frac{1}{n}{({\sum }_{i=1}^n{S}_i^{\prime })}^2}}?$$

Yule empirically observed the distribution of this statistic to be heavily dispersed and frequently large in absolute value, leading him to call it “nonsense correlation.” This unexpected finding led to his formulation of two concrete questions, each of which would remain open for more than ninety years: (i) Find (analytically) the variance of ${\mathit{\rho }}_n$ as $n\to \mathrm{\infty }$ and (ii): Find (analytically) the higher order moments and the density of ${\mathit{\rho }}_n$ as $n\to \mathrm{\infty }$. In 2017, Ernst, Shepp and Wyner (Ann. Statist. 45 (2017) 1789–1809) considered the empirical correlation coefficient

$$\mathit{\rho }:=\frac{{\int }_0^1{W}_1(t)W_2(t)dt-{\int }_0^1{W}_1(t)dt{\int }_0^1{W}_2(t)dt}{\sqrt{{\int }_0^1{W}_1^2(t)dt-{\left({\int }_0^1{W}_1(t)dt\right)}^2}\sqrt{{\int }_0^1{W}_2^2(t)dt-{\left({\int }_0^1{W}_2(t)dt\right)}^2}}$$

of two independent Wiener processes $W_1,W_2$, the limit to which ${\mathit{\rho }}_n$ converges weakly, as was first shown by P.C.B. Phillips (J. Econometrics 33 (1986) 311–340). Using tools from integral equation theory, Ernst, Shepp and Wyner (Ann. Statist. 45 (2017) 1789–1809) closed question (i) by explicitly calculating the second moment of ρ to be .240522. This paper adopts a completely different approach to the same question, rooted in an earlier literature on the laws of quadratic functionals of Gaussian diffusions (in particular, (Adv. in Appl. Probab. 25 (1993) 570–584; Stoch. Stoch. Rep. 41 (1992) 201–218)). This allows us to develop an Itô-formula approach from which we calculate expressions for the Laplace transform of ρ, leading to expressions for the moments which we evaluate up to order 16, thereby closing question (ii). This leads, for the first time, to an approximation to the density of Yule’s nonsense correlation. The broad applicability of this approach is demonstrated by answering the corresponding questions when the pair of independent Brownian motions is replaced by a pair of correlated Brownian motions, or by two independent Ornstein-Uhlenbeck processes, or by two independent Brownian bridges. We conclude by extending the definition of ρ to the time interval $[0,T]$ for any $T>0$ and prove a Central Limit Theorem for the case of two independent Ornstein-Uhlenbeck processes.

This work resolves the following question in non-Euclidean statistics: Is it possible to consistently estimate the Fréchet mean set of an unknown population distribution, with respect to the Hausdorff metric, when given access to independent identically-distributed samples? Our affirmative answer is based on a careful analysis of the “relaxed empirical Fréchet mean set estimators” which identify the set of near-minimizers of the empirical Fréchet functional and where the amount of “relaxation” vanishes as the number of data tends to infinity. On the theoretical side, our results include exact descriptions of which relaxation rates give weak consistency and which give strong consistency, as well as a description of an estimator which (assuming only the finiteness of certain moments and a mild condition on the metric entropy of the underlying metric space) adaptively finds the fastest possible relaxation rate for strongly consistent estimation. On the applied side, we consider the problem of estimating the set of Fermat-Weber points of an unknown distribution in the space of equidistant trees endowed with the tropical projective metric; in this setting, we provide an algorithm that provably implements our adaptive estimator, and we apply this method to real phylogenetic data.

We introduce a random recursive tree model with two communities, called balanced community modulated random recursive tree, or BCMRT in short. In this setting, pairs of nodes of different type appear sequentially. Each node of the pair decides independently to attach to their own type with probability $1-q$, or to the other type with probability q, and then chooses its parent uniformly within the set of existing nodes with the selected type. We find that the limiting degree distributions coincide for different q. Therefore, as far as inference is concerned, other statistics have to be studied. We first consider the setting where the time-labels of the nodes, i.e., their time of arrival, are observed but their type is not. In this setting, we design a consistent estimator for q and provide bounds for the feasibility of testing between two different values of q. Moreover, we show that if q is small enough, then it is possible to cluster the nodes in a way correlated with the true partition, even though the algorithm is exponential in time (in passing, we show that our clustering procedure is intimately connected to the NP-hard problem of minimum fair bisection). In the unlabelled setting, i.e., when only the tree structure is observed, we show that it is possible to test between different values of q in a strictly better way than by random guessing. This follows from a delicate analysis of the sum-of-distances statistic.

We consider a pure-jump stable Cox-Ingersoll-Ross (α-stable CIR) process driven by a non-symmetric stable Lévy process with jump activity $\mathit{\alpha }\in (1,2)$ and we address the joint estimation of drift, scaling and jump activity parameters from high-frequency observations of the process on a fixed time period. We first prove the existence of a consistent, rate optimal and asymptotically conditionally Gaussian estimator based on an approximation of the likelihood function. Moreover, uniqueness of the drift estimators is established assuming that the scaling coefficient and the jump activity are known or consistently estimated. Next we propose easy-to-implement preliminary estimators of all parameters and we improve them by a one-step procedure.

Enriching Brownian motion with regenerations from a fixed regeneration distribution μ at a particular regeneration rate κ results in a Markov process that has a target distribution π as its invariant distribution. For the purpose of Monte Carlo inference, implementing such a scheme requires firstly selection of regeneration distribution μ, and secondly computation of a specific constant C. Both of these tasks can be very difficult in practice for good performance. We introduce a method for adapting the regeneration distribution, by adding point masses to it. This allows the process to be simulated with as few regenerations as possible and obviates the need to find said constant C. Moreover, the choice of fixed μ is replaced with the choice of the initial regeneration distribution, which is considerably less difficult. We establish convergence of this resulting self-reinforcing process and explore its effectiveness at sampling from a number of target distributions. The examples show that adapting the regeneration distribution guards against poor choices of fixed regeneration distribution and can reduce the error of Monte Carlo estimates of expectations of interest, especially when π is skewed.

A limit theorem for the largest interpoint distance of p independent and identically distributed points in ${\mathbb{R}}^n$ to the Gumbel distribution is proved, where the number of points $p=p_n$ tends to infinity as the dimension of the points $n\to \mathrm{\infty }$. The theorem holds under moment assumptions and corresponding conditions on the growth rate of p. We obtain a plethora of ancillary results such as the joint convergence of maximum and minimum interpoint distances. Using the inherent sum structure of interpoint distances, our result is generalized to maxima of dependent random walks with non-decaying correlations and we also derive point process convergence. An application of the maximum interpoint distance to testing the equality of means for high-dimensional random vectors is presented. Moreover, we study the largest off-diagonal entry of a sample covariance matrix. The proofs are based on the Chen-Stein Poisson approximation method and Gaussian approximation to large deviation probabilities.

We provide an elementary proof of geometric synchronisation for scalar conservation laws on a domain with Dirichlet boundary conditions. Unlike previous results, our proof does not rely on a strict maximum principle, and builds instead on a quantitative estimate of the dissipation at the boundary. We identify a coercivity condition under which the estimates are uniform over all initial conditions, via the construction of suitable super- and sub-solutions. In lack of such coercivity our results build on $L^p$ energy estimates and a Lyapunov structure.

We propose a general framework for modelling network data that is designed to describe aspects of non-exchangeability, with an explicit parameter describing the degree of non-exchangeability. Conditional on latent (unobserved) variables, the edges of the network are generated by their finite growth history (via a latent order) while the marginal probabilities of the adjacency matrix are modeled by a generalization of a graph limit function (or a graphon). In particular, we study the estimation, clustering and degree behavior of the network in this setting. We determine (i) the least squares estimator of a composite graphon attaining the minimax rate under weak dependence with respect to squared error loss; (ii) that spectral clustering is able to consistently detect the latent membership when the block-wise constant composite graphon is considered under additional conditions; and (iii) we are able to construct models with heavy-tailed empirical degrees under specific scenarios and parameter choices. We find conditions under which the spectral clustering is consistent under non-exchangeability, revealing that the application scope of classification can be broader than classic $i.i.d$. or exchangeable assumptions. In aggregate, we explore why and under which general conditions non-exchangeable network data can be described by a stochastic block model. The new modelling framework is able to capture empirically important characteristics of network data such as sparsity combined with heavy tailed degree distribution, and add understanding as to what generative mechanisms will make them arise.

We describe the solution of an optimal stopping problem for a stable Lévy process killed at state-dependent rate. The killing rate is chosen in such a way that the killed process remains self-similar, and the solution to the optimal stopping problem is obtained by characterising a self-similar Markov process associated with the stable process. The optimal stopping strategy is to stop upon first passage into an interval, found explicitly in terms of the parameters of the model.

Consider a critical branching Lévy process $\{X_t,t\ge 0\}$ with branching rate $\mathit{\beta }>0$, offspring distribution $\{p_k:k\ge 0\}$ and spatial motion $\{{\mathrm{\xi }}_t,{\mathcal{P}}_x\}$. For any $t\ge 0$, let $N_t$ be the collection of particles alive at time t, and, for any $u\in N_t$, let $X_u(t)$ be the position of u at time t. We study the tail probability of the maximal displacement $M:={sup}_{t>0}{sup}_{u\in N_t}{X}_u(t)$ under the assumption ${lim}_{n\to \mathrm{\infty }}{n}^{\mathit{\alpha }}{\sum }_{k=n}^{\mathrm{\infty }}{p}_k=\mathrm{\kappa }\in (0,\mathrm{\infty })$ for some $\mathit{\alpha }\in (1,2)$, ${\mathcal{E}}_0({\mathrm{\xi }}_1)=0$ and ${\mathcal{E}}_0({({\mathrm{\xi }}_1^{+})}^r)\in (0,\mathrm{\infty })$ for some $r>2\mathit{\alpha }∕(\mathit{\alpha }-1)$. Our main result is a generalization of the main result of Sawyer and Fleischman (1979) for branching Brownian motions and that of Lalley and Shao (2015) for branching random walks, both of these results are proved under the assumption ${\sum }_{k=0}^{\mathrm{\infty }}{k}^3{p}_k<\mathrm{\infty }$.

Principal component analysis has been one of prominent techniques in multi-dimensional statistics, and the core is to estimate the eigenvalues in descending ordering with their corresponding eigenvectors. However, when data are collected longitudinally in many scientific applications, the eigenvalues become dynamic over time, and the ordering of them may not be well-defined, for instance, does not coincide with the pointwise ordering. To deal with this issue, we propose a new framework, namely the dynamic principal component analysis. This addresses the identifiability of principal components from a global perspective, and transforms the problem into a regression model for data situated on the orthogonal matrix group space. The one-step unrolling method is exploited to solve the regression problem with a suitably constructed regular base curve. The minimax rate of the proposed estimators is established through theoretical analysis of the one-step unrolling and its connection to smoothing spline in the context of manifold-valued data.

Let ${\mathbf{B}}_n=n^{-1}({\mathbf{R}}_n+{\mathbf{T}}_n^{1∕2}{\mathbf{X}}_n){({\mathbf{R}}_n+{\mathbf{T}}_n^{1∕2}{\mathbf{X}}_n)}^{\ast }$, where ${\mathbf{X}}_n$ is a $p\times n$ matrix with independent standardized random variables, ${\mathbf{R}}_n$ is a $p\times n$ non-random matrix and ${\mathbf{T}}_n$ is a $p\times p$ non-random, nonnegative definite Hermitian matrix. The matrix ${\mathbf{B}}_n$ is referred to as the information-plus-noise type matrix, where ${\mathbf{R}}_n$ contains the information and ${\mathbf{T}}_n^{1∕2}{\mathbf{X}}_n$ is the noise matrix with the covariance matrix ${\mathbf{T}}_n$. It is known that, as $n\to \mathrm{\infty }$, if $p∕n$ converges to a positive number, the empirical spectral distribution of ${\mathbf{B}}_n$ converges almost surely to a nonrandom limit, under some conditions. In this paper, we prove that, under certain conditions on the eigenvalues of ${\mathbf{R}}_n$ and ${\mathbf{T}}_n$, for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all n sufficiently large.

In this paper, based on the techniques of Malliavin calculus, we provide some new concentration inequalities for the running supremum of the Itô stochastic integral with unbounded integrands. Several applications and examples are provided as well.

We propose a new model for non-stationary integer-valued time series which is particularly suitable for data with a strong trend. In contrast to popular Poisson-INGARCH models, but in line with classical GARCH models, we propose to pick the conditional distributions from nearly scale invariant families where the mean absolute value and the standard deviation are of the same order of magnitude. As an important prerequisite for applications in statistics, we prove absolute regularity of the count process with exponentially decaying coefficients.

Both locally stationary processes and irregular models have had a long story of success in statistics and time series analysis. We combine both concepts and consider a nonparametric, first-order autoregressive model with irregular, positive innovations, where we assume that the coefficient function is Hölder continuous and positive. To estimate this function, we use a quasi-maximum likelihood based approach. A precise control of this method demands a delicate analysis of extremes of certain weakly dependent processes, our main result being a concentration inequality for such quantities. Based on our analysis, upper and matching minimax lower bounds are derived, showing the optimality of our estimators. Unlike the regular case, the information theoretic complexity depends both on the smoothness and an additional shape parameter, characterizing the irregularity of the underlying distribution. The results and ideas for the proofs are very different from classical and more recent methods in connection with locally stationary processes.

In this paper, we establish quadratic transportation cost inequalities for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise on the whole real line $\mathbb{R}$. Since the space variable is defined on the unbounded domain $\mathbb{R}$, the inequalities are proved under a weighted $L^2$-norm and a weighted uniform metric in the so-called $L_{tem}^2$, $C_{tem}$ spaces. The new moments estimates of the stochastic convolution with respect to space-time white noise play an important role. In addition, the transportation cost inequalities are also obtained for the stochastic reaction diffusion equations with random initial values.

In this paper we focus on the problem of the parameter estimation for a class of systems described by multi-scale McKean-Vlasov stochastic differential equations (MVSDEs, for short) with small noise, where the coefficients depend on the slow component, the fast component, and their own distributions. Firstly, we prove an optimal strong convergence rate of the strong averaging principle of the slow process by using the technique of Poisson equation. Secondly, we construct the maximum likelihood estimator (MLE, for short) and prove its consistency and asymptotic normality. Finally, an example is presented to illustrate the theoretical findings.

We study empirical variants of the halfspace (Tukey) depth of a probability measure μ, which are obtained by replacing μ with the corresponding weighted empirical measure. We prove analogues of the Marcinkiewicz–Zygmund strong law of large numbers and of the law of the iterated logarithm in terms of set inclusions and for the Hausdorff distance between the theoretical and empirical variants of depth trimmed regions. In the special case of μ being the uniform distribution on a convex body K, the depth trimmed regions are convex floating bodies of K, and we obtain strong limit theorems for their empirical estimators.