It is known that the membership in a given reproducing kernel Hilbert space (RKHS) of the samples of a Gaussian process X is controlled by a certain nuclear dominance condition. However, it is less clear how to identify a “small” set of functions (not necessarily a vector space) that contains the samples. This article presents a general approach for identifying such sets. We use scaled RKHSs, which can be viewed as a generalisation of Hilbert scales, to define the sample support set as the largest set which is contained in every element of full measure under the law of X in the σ-algebra induced by the collection of scaled RKHS. This potentially non-measurable set is then shown to consist of those functions that can be expanded in terms of an orthonormal basis of the RKHS of the covariance kernel of X and have their squared basis coefficients bounded away from zero and infinity, a result suggested by the Karhunen–Loève theorem.

This article illustrates how indirect or prior information can be optimally used to construct a prediction region that maintains a target frequentist coverage rate. If the indirect information is accurate, the volume of the prediction region is lower on average than that of other regions with the same coverage rate. Even if the indirect information is inaccurate, the resulting region still maintains the target coverage rate. Such a prediction region can be constructed for models that have a complete sufficient statistic, which includes many widely-used parametric and nonparametric models. Particular examples include a Bayes-optimal conformal prediction procedure that maintains a constant coverage rate across distributions in a nonparametric model, as well as a prediction procedure for the normal linear regression model that can utilize a regularizing prior distribution, yet maintain a frequentist coverage rate that is constant as a function of the model parameters and explanatory variables. No results in this article rely on asymptotic approximations.

We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and π-canonical kernels, we show that we can recover the convergence rate of Arcones and Giné who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernstein’s type inequality.

Assuming further that the Markov chain starts from its invariant distribution, we prove a Bernstein-type concentration inequality that provides sharper convergence rate for small variance terms.

Determinantal point processes (DPPs) are repulsive point processes where the interaction between points depends on the determinant of a positive-semi definite matrix.

In this paper, we study the limiting process of L-ensembles based on kernel matrices, when the kernel function becomes flat (so that every point interacts with every other point, in a sense). We show that these limiting processes are best described in the formalism of extended L-ensembles and partial projection DPPs, and the exact limit depends mostly on the smoothness of the kernel function. In some cases, the limiting process is even universal, meaning that it does not depend on specifics of the kernel function, but only on its degree of smoothness.

Since flat-limit DPPs are still repulsive processes, this implies that practically useful families of DPPs exist that do not require a spatial length-scale parameter.

A high-dimensional sample correlation matrix is an important random matrix in multivariate statistical analysis. Its central limit theory is one of the main theoretical bases for making statistical inferences on high-dimensional correlation matrices. Under the high-dimensional framework in which the data dimension tends to infinity proportionally with the sample size, we establish the central limit theorems (CLT) for the linear spectral statistics (LSS) of sample correlation matrices in two settings: (1) the population follows an independent component structure; (2) the population follows an elliptical structure, including some heavy-tailed distributions. The results show that the CLTs of the LSS of the sample correlation matrices are very different in the two settings. In particular, even if the population correlation matrix is an identity matrix, the CLTs are different in the two settings. An application of our two established CLTs is provided.

In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree, satisfying Kesten-Stigum condition. We study the large deviations of the extremal process, formed by the appropriately normalized positions in the n-th generation and show that the large extreme-positions form clusters in the limit. As a consequence of this, we also study the large deviations of the maximum among positions at the n-th generation.

Under Lyapunov and monotone conditions, the exponential ergodicity in the induced Wasserstein quasi-distance is proved for a class of non-dissipative McKean-Vlasov SDEs, which strengthen some recent results established under dissipative conditions in long distance. Moreover, when the SDE is order-preserving, the exponential ergodicity is derived in the Wasserstein distance induced by one-dimensional increasing functions chosen according to the coefficients of the equation.

We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green function comparison along a continuous interpolating matrix flow for a long time. Less precise estimates are also obtained in the left tail.

Bayesian posterior distributions are widely used for inference, but their dependence on a statistical model creates some challenges. In particular, there may be lots of nuisance parameters that require prior distributions and posterior computations, plus a potentially serious risk of model misspecification bias. Gibbs posterior distributions, on the other hand, offer direct, principled, probabilistic inference on quantities of interest through a loss function, not a model-based likelihood. Here we provide simple sufficient conditions for establishing Gibbs posterior concentration rates when the loss function is of a sub-exponential type. We apply these general results in a range of practically relevant examples, including mean regression, quantile regression, and sparse high-dimensional classification. We also apply these techniques in an important problem in medical statistics, namely, estimation of a personalized minimum clinically important difference.

Randomization tests are based on a re-randomization of existing data to gain data-dependent critical values that lead to exact hypothesis tests under special circumstances. However, it is not always possible to re-randomize data in accordance with the physical randomization from which the data have been obtained. As a consequence, statistical tests typically cannot control the type I error probability if the available sample is not overly large. Still, similarly as the bootstrap, data re-randomization can be used to improve the type I error control. However, in contrast to bootstrap and permutation test theory, a general asymptotic theory under weak null hypotheses has not yet been developed for randomization tests. It is thus the aim of this paper to provide a conveniently applicable theory on the asymptotic validity of randomization tests with test statistics that are asymptotically normal under weak null hypotheses. This can also be used to justify the asymptotic reliability of randomization-based confidence intervals.

To achieve this, the present article creates a link between two well-established fields in mathematical statistics: empirical processes and inference based on randomization via algebraic groups. To this end, a broadly applicable conditional weak convergence theorem is developed for empirical processes that are based on randomized observations. Random elements of an algebraic group are applied to the data vectors from which the randomized version of a statistic is derived. Combining a variant of the functional delta-method with a suitable studentization of the statistic, asymptotically exact hypothesis tests can be deduced, while the finite sample exactness property under group-invariant sub-hypotheses is preserved. The methodology is exemplified with three examples: the Pearson correlation coefficient, a Mann-Whitney effect based on right-censored paired data, and a competing risks analysis. The practical usefulness of the approaches is assessed through simulation studies and an application to data from patients suffering from diabetic retinopathy.

In this paper, we establish the almost sure convergence of two-timescale stochastic gradient descent algorithms in continuous time under general noise and stability conditions, extending well known results in discrete time. We analyse algorithms with additive noise and those with non-additive noise. In the non-additive case, our analysis is carried out under the assumption that the noise is a continuous-time Markov process, controlled by the algorithm states. The algorithms we consider can be applied to a broad class of bilevel optimisation problems. We study one such problem in detail, namely, the problem of joint online parameter estimation and optimal sensor placement for a partially observed diffusion process. We demonstrate how this can be formulated as a bilevel optimisation problem, and propose a solution in the form of a continuous-time, two-timescale, stochastic gradient descent algorithm. Furthermore, under suitable conditions on the latent signal, the filter, and the filter derivatives, we establish almost sure convergence of the online parameter estimates and optimal sensor placements to the stationary points of the asymptotic log-likelihood and asymptotic filter covariance, respectively. We also provide numerical examples, illustrating the application of the proposed methodology to a partially observed Beneš equation, and a partially observed stochastic advection-diffusion equation.

Fisher (1943) claimed that the expected value of the sample variance of the number of species found in large samples, each of n specimens taken from the same population, is asymptotically $\mathit{\theta }log2$. This is at odds with the value $\mathit{\theta }logn$ obtained directly from the Ewens Sampling Formula (ESF), where θ specifies the rate at which new species are found. To resolve this apparent contradiction, we assume the species frequency spectrum in the population is determined by the ESF and that the samples are disjoint subsets drawn sequentially from this single population. We find an explicit formula for the required expected value for p samples of arbitrary size; in the limit of large equally-sized samples, it indeed has the value $\mathit{\theta }log2$. We obtain limit theorems for the sample variance of p samples of size n under various limiting regimes as $p,n$ or both tend to ∞. We discuss further the behavior of the number of species present in all samples, and revisit Fisher’s log-series distribution as the limiting distribution of the number of specimens observed in typical species in a future, large sample.

We study Bayesian estimation of finite mixture models in a general setup where the number of components is unknown and allowed to grow with the sample size. An assumption on growing number of components is a natural one as the degree of heterogeneity present in the sample can grow and new components can arise as sample size increases, allowing full flexibility in modeling the complexity of data. This however will lead to a high-dimensional model which poses great challenges for estimation. We novelly employ the idea of a sample size dependent prior in a Bayesian model and establish a number of important theoretical results. We first show that under mild conditions on the prior, the posterior distribution concentrates around the true mixing distribution at a near optimal rate with respect to the Wasserstein distance. Under a separation condition on the true mixing distribution, we further show that a better and adaptive convergence rate can be achieved, and the number of components can be consistently estimated. Furthermore, we derive optimal convergence rates for the higher-order mixture models where the number of components diverges arbitrarily fast. In addition, we suggest a simple recipe for using Dirichlet process (DP) mixture prior for estimating the finite mixture models and provide theoretical guarantees. In particular, we provide a novel solution for adopting the number of clusters in a DP mixture model as an estimate of the number of components in a finite mixture model. Simulation study and real data applications are carried out demonstrating the utilities of our method.

Suppose that the (normalised) partial sum of a stationary sequence converges to a standard normal random variable. Given sufficiently moments, when do we have a rate of convergence of $n^{-1∕2}$ in the uniform metric, in other words, when do we have the optimal Berry-Esseen bound? We study this question in a quite general framework and find the (almost) sharp dependence conditions. The result applies to many different processes and dynamical systems. As specific, prominent examples, we study functions of the doubling map 2x mod 1, the left random walk on the general linear group and functions of linear processes.

A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter $p\in [0,1]$ means the algorithm does not know p, but has sample access to independent draws of a Bernoulli random variable with mean equal to p. In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector $x\in \mathcal{P}$ for a given polytope $\mathcal{P}\subset {[0,1]}^n$, output a randomized vertex such that the expected value of the i-th coordinate is exactly equal to $x_i$. For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix $[x_{ij}]$ and asked to sample a matching such that the probability of each edge $(i,j)$ be present in the matching is exactly equal to $x_{ij}$.

We show that a polytope $\mathcal{P}$ admits a Bernoulli factory if and and only if $\mathcal{P}$ is the intersection of ${[0,1]}^n$ with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on $\mathcal{P}$. The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings.

We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the k-uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.11

For a generalization of Johnstone’s spiked model, a covariance matrix with eigenvalues all one but M of them, the number of features N comparable to the number of samples $n:N=N(n),M=M(n),{\mathit{\gamma }}^{-1}\le \frac{N}{n}\le \mathit{\gamma }$ where $\mathit{\gamma }\in (0,\mathrm{\infty })$, we obtain consistency rates in the form of CLTs for separated spikes tending to infinity fast enough whenever M grows slightly slower than $n:{lim}_{n\to \mathrm{\infty }}\frac{\sqrt{logn}}{log\frac{n}{M(n)}}=0$. Our results fill a gap in the existing literature in which the largest range covered for the number of spikes has been $o(n^{1∕6})$ and reveal a certain degree of flexibility for the centering in these CLTs inasmuch as it can be empirical, deterministic, or a sum of both. Furthermore, we derive consistency rates of their corresponding empirical eigenvectors to their true counterparts, which turn out to depend on the relative growth of these eigenvalues.

We study stability of the sharp Poincaré constant of the invariant probability measure of a reversible diffusion process satisfying some natural conditions. The proof is based on the spectral interpretation of Poincaré inequalities and Stein’s method. In particular, these results are applied to gamma distributions, to the Brownian motion on spheres and to the Brascamp-Lieb inequality for one-dimensional log-concave measures.

Let $X=(X_1,X_2,\dots )$ be a sequence of random variables with values in a standard space $(S,\mathcal{B})$. Suppose

$$X_1\sim \mathrm{\nu }\text{and}P(X_{n+1}\in \cdot \mid X_1,\dots ,X_n)=\frac{\mathit{\theta }\mathrm{\nu }(\cdot )+{\sum }_{i=1}^{n}K(X_i)(\cdot )}{n+\mathit{\theta }}\text{a.s.}$$

where $\mathit{\theta }>0$ is a constant, ν a probability measure on $\mathcal{B}$, and K a random probability measure on $\mathcal{B}$. Then, X is exchangeable whenever K is a regular conditional distribution for ν given any sub-σ-field of $\mathcal{B}$. Under this assumption, X enjoys all the main properties of classical Dirichlet sequences, including Sethuraman’s representation, conjugacy property, and convergence in total variation of predictive distributions. If μ is the weak limit of the empirical measures, conditions for μ to be a.s. discrete, or a.s. non-atomic, or $\mathrm{\mu }\ll \mathrm{\nu }$ a.s., are provided. Two CLT’s are proved as well. The first deals with stable convergence while the second concerns total variation distance.

We consider a symmetric finite-range contact process on $\mathbb{Z}$ with two types of particles (or infections), which propagate according to the same supercritical rate and die (or heal) at rate 1. Particles of type 1 can enter any site in $(-\mathrm{\infty },0]$ that is empty or occupied by a particle of type 2 and, analogously, particles of type 2 can enter any site in $[1,\mathrm{\infty })$ that is empty or occupied by a particle of type 1. Also, at most one particle can occupy each site. We prove that the process with initial configuration ${\mathbb{1}}_{(-\mathrm{\infty },0]}+{\mathbb{21}}_{[1,\mathrm{\infty })}$ converges in distribution to an invariant measure different from the non trivial invariant measure of the classic contact process. In addition, we prove that for any initial configuration the process converges to a convex combination of four invariant measures.

High-dimensional feature vectors are likely to contain sets of measurements that are approximate replicates of one another. In complex applications, or automated data collection, these feature sets are not known a priori, and need to be determined.

This work proposes a class of latent factor models on the observed, high-dimensional, random vector $X\in {\mathbb{R}}^p$, for defining, identifying and estimating the index set of its approximately replicate components. The model class is parametrized by a $p\times K$ loading matrix A that contains a hidden sub-matrix whose rows can be partitioned into groups of parallel vectors. Under this model class, a set of approximate replicate components of X corresponds to a set of parallel rows in A: these entries of X are, up to scale and additive error, the same linear combination of the K latent factors; the value of K is itself unknown.

The problem of finding approximate replicates in X reduces to identifying, and estimating, the location of the hidden sub-matrix within A, and of the partition $\mathcal{H}$ of its row index set H. Both H and $\mathcal{H}$ can be fully characterized in terms of a new family of criteria based on the correlation matrix of X, and their identifiability, as well as that of the unknown latent dimension K, are obtained as consequences. The constructive nature of the identifiability arguments enables computationally efficient procedures, with consistency guarantees.

Furthermore, when the loading matrix A has a particular sparse structure, provided by the errors-in-variable parametrization, the difficulty of the problem is elevated. The task becomes that of separating out groups of parallel rows that are proportional to canonical basis vectors from other, possibly dense, parallel rows in A. This is met under a scale assumption, via a principled way of selecting the target row indices, guided by the successive maximization of Schur complements of appropriate covariance matrices. The resulting procedure is an enhanced version of that developed for recovering general parallel rows in A. It is also computationally efficient, consistent. It has immediate applications to latent space overlapping clustering and the estimation of loading matrices that satisfy a canonical parametrization.

We consider discrete probability laws on the real line, whose characteristic functions are separated from zero. This class includes arbitrary discrete infinitely divisible laws and lattice probability laws, whose characteristic functions have no zeroes on the real line. We show that characteristic functions of such laws admit spectral Lévy–Khinchine type representation with non-monotonic Lévy spectral function. We also apply the representations of such laws to obtain limit and compactness theorems with convergence in variation to probability laws from this class.

We study the distribution of the unobserved states of two measure-valued diffusions of Fleming–Viot and Dawson–Watanabe type, conditional on observations from the underlying populations collected at past, present and future times. If seen as nonparametric hidden Markov models, this amounts to finding the smoothing distributions of these processes, which we show can be explicitly described in recursive form as finite mixtures of laws of Dirichlet and gamma random measures respectively. We characterize the time-dependent weights of these mixtures, accounting for potentially different time intervals between data collection times, and fully describe the implications of assuming a discrete or a nonatomic distribution for the underlying process that drives mutations. In particular, we show that with a nonatomic mutation offspring distribution, the inference automatically upweights mixture components that carry, as atoms, observed types shared at different collection times. The predictive distributions for further samples from the population conditional on the data are also identified and shown to be mixtures of generalized Pólya urns, conditionally on a latent variable in the Dawson–Watanabe case.

In the paper we prove a lower bound for subsolutions of the integro-differential equation: $-Au+cu=0$ in a domain D. It states that there exists a Borel function ψ, strictly positive on D, depending only on the coefficients of the operator A, c and D such that for any subsolution $u(\cdot )$, that satisfies ${sup}_{y\in D_S}u(y)\ge 0$, one can find a constant $a>0$ (that in general depends on u), for which ${sup}_{y\in D_S}u(y)-u(x)\ge a\mathit{\psi }(x)$, $x\in D$. The bound is valid for a wide class of Lévy type integro-differential operators A, non-negative, bounded and measurable function c and a quite general domain $D\subset {\mathbb{R}}^d$. Here $D_S$ is a certain set containing the closure of D and determined by the support of the Levy jump measure associated with A. In some cases a non-negative eigenfunction corresponding to the operator in D can be admitted as the function ψ. In particular, this occurs when the transition probability semigroup associated with A is ultracontractive. The main assumptions made about A are: there exists a strong Markov solution to the martingale problem associated with the operator and its resolvent satisfies some minorization condition. This type of a result we call the generalized Hopf lemma.

Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into d-dimensional Euclidean spaces, ${\mathbb{R}}^d$, have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggests that the problem is challenging. This paper provides extension theorems for multiradial characteristic functions that are defined in balls embedded in ${\mathbb{R}}^d$ cross, either ${\mathbb{R}}^{d^{\prime }}$ or the unit sphere ${\mathbb{S}}^{d^{\prime }}$ embedded in ${\mathbb{R}}^{d^{\prime }+1}$, for any two positive integers d and $d^{\prime }$. We then examine Turning Bands operators that provide bijections between the class of multiradial correlation functions in given product spaces, and multiradial correlations in product spaces having different dimensions. The combination of extension theorems with Turning Bands provides a connection with random fields that are defined in balls cross linear or circular time.

We consider two Gaussian measures $\mathrm{\mu },\tilde{\mathrm{\mu }}$ on a separable Hilbert space, with fractional-order covariance operators $A^{-2\mathit{\beta }}$ and ${\tilde{A}}^{-2\tilde{\mathit{\beta }}}$, respectively, and derive necessary and sufficient conditions on $A,\tilde{A}$ and $\mathit{\beta },\tilde{\mathit{\beta }}>0$ for I. equivalence of the measures μ and $\tilde{\mathrm{\mu }}$, and II. uniform asymptotic optimality of linear predictions for μ based on the misspecified measure $\tilde{\mathrm{\mu }}$. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle–Matérn Gaussian random fields, where A and $\tilde{A}$ are elliptic second-order differential operators, formulated on a bounded Euclidean domain $\mathcal{D}\subset {\mathbb{R}}^d$ and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle–Matérn fields.

We study a functional version of nonstationary fractionally integrated time series, covering the functional unit root as a special case. The time series taking values in an infinite-dimensional separable Hilbert space are projected onto a finite number of sub-spaces, the level of nonstationarity allowed to vary over them. Under regularity conditions, we derive a weak convergence result for the projection of the fractionally integrated functional process onto the asymptotically dominant sub-space, which retains most of the sample information carried by the original functional time series. Through the classic functional principal component analysis of the sample variance operator, we obtain the eigenvalues and eigenfunctions which span a sample version of the dominant sub-space. Furthermore, we introduce a simple ratio criterion to consistently estimate the dimension of the dominant sub-space, and use a semiparametric local Whittle method to estimate the memory parameter. Monte-Carlo simulation studies are given to examine the finite-sample performance of the developed techniques.

This article investigates whether time-varying quantile regression curves are the same up to the horizontal shift or not. The errors and the covariates involved in the regression model are allowed to be locally stationary. We formalize this issue in a corresponding non-parametric hypothesis testing problem, and develop an integrated-squared-norm based test (SIT) as well as a simultaneous confidence band (SCB) approach. The asymptotic properties of SIT and SCB under null and local alternatives are derived. Moreover, the asymptotic properties of these tests are also studied when the compared data sets are dependent. We then propose valid wild bootstrap algorithms to implement SIT and SCB. Furthermore, the usefulness of the proposed methodology is illustrated via analysing simulated and real data related to COVID-19 outbreak.

We consider uniform spanning tree (UST) in topological polygons with $2N$ marked points on the boundary with alternating boundary conditions. In an earlier work by Liu-Peltola-Wu, the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE${}_8$. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE${}_2$. The conclusion is a generalization of an earlier work by Han-Liu-Wu where the authors derive the scaling limit of the LERW branch of UST when $N=2$. When $N=2$, the limiting law is SLE${}_2(-1,-1;-1,-1)$. However, the limiting law is no longer in the family of SLE${}_2(\mathit{\rho })$ process as long as $N\ge 3$.

We investigate the Rényi entropy of sums of independent integer-valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the Rényi entropy for sums of independent Bernoulli random variables. As applications, we prove that a discrete “min-entropy power” is superadditive with respect to convolution modulo a universal constant, and give new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the “Poisson regime”.

The main purpose of the paper is to investigate the possibility of applying the Stein-Chen approach to estimate the ${\mathit{\chi }}^2$ distance between Poisson distribution and Poisson binomial distribution. Earlier results concerning ${\mathit{\chi }}^2$ distance between the above mentioned distributions either used analytical approach heavily based on the analysis of the generating functions or on rather lengthy and complicated elementary calculations. Applying the Stein-Chen approach we succeed in providing a very quick proof of upper bounds for ${\mathit{\chi }}^2$ distance that are of comparable strength to the earlier estimates obtained by other approaches.

We study random normal matrix models whose eigenvalues tend to be distributed within a narrow “band” around the unit circle of width proportional to $1∕n$, where n is the size of matrices. For general radially symmetric potentials with various boundary conditions, we derive the scaling limits of the correlation functions, some of which appear in the previous literature notably in the context of almost-Hermitian random matrices. We also obtain that fluctuations of the maximal and minimal modulus of the ensembles follow the Gumbel or exponential law depending on the boundary conditions.

Let ${\mathrm{\xi }}_1$, ${\mathrm{\xi }}_2,\dots$ be i.i.d. random variables of zero mean and finite variance and ${\mathrm{\eta }}_1$, ${\mathrm{\eta }}_2,\dots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an α-stable distribution, $\mathit{\alpha }\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump ${\mathrm{\xi }}_k$ occurs; if the present position of the Markov chain is nonpositive, then the jump ${\mathrm{\eta }}_k$ occurs. We prove functional limit theorems for this and two closely related Markov chains under Donsker’s scaling. The weak limit is a nonnegative process ${(X(t))}_{t\ge 0}$ satisfying a stochastic equation $\mathrm{d}X(t)=\mathrm{d}W(t)+\mathrm{d}{U}_{\mathit{\alpha }}(L_X^{(0)}(t))$, where W is a Brownian motion, $U_{\mathit{\alpha }}$ is an α-stable subordinator which is independent of W, and $L_X^{(0)}$ is a local time of X at 0. Also, we explain that X is a Feller Brownian motion with a ‘jump-type’ exit from 0.

A regression problem with dependent data is considered. Regularity assumptions on the dependency of the data are introduced, and it is shown that under suitable structural assumptions on the regression function a deep recurrent neural network estimate is able to circumvent the curse of dimensionality.

Let ${Mat}_{\mathbb{C}}(K,N)$ be the space of $K\times N$ complex matrices. Let ${\mathbf{B}}_t$ be Brownian motion on ${Mat}_{\mathbb{C}}(K,N)$ starting from the zero matrix and $\mathbf{M}\in {Mat}_{\mathbb{C}}(K,N)$. We prove that, with $K\ge N$, the N eigenvalues of ${\left({\mathbf{B}}_t+t\mathbf{M}\right)}^{\ast }\left({\mathbf{B}}_t+t\mathbf{M}\right)$ form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman (Ann. Probab. 9 (1981) 573–582) for multidimensional Brownian motion with drift which corresponds to $N=1$. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe (Z. Wahrsch. Verw. Gebiete 31 (1975) 115–124) and studied by Pitman and Yor (In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980) (1981) 285–370 Springer), conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.

We study the weighted Poincaré constant $C(p,w)$ of a probability density p with weight function w using integration methods inspired by Stein’s method. We obtain a new version of the Chen-Wang variational formula which, as a byproduct, yields simple upper and lower bounds on $C(p,w)$ in terms of the so-called Stein kernel of p. We also iterate these variational formulas so as to build sequences of nested intervals containing the Poincaré constant, sequences of functions converging to said constant, as well as sequences of functions converging to the solutions of the corresponding spectral problem. Our results rely on the properties of a pseudo inverse operator of the classical Sturm-Liouville operator. We illustrate our methods on a variety of examples: Gaussian functionals, weighted Gaussian, beta, gamma, Subbotin, and Weibull distributions.