The mixed-membership stochastic block model (MMSBM) is a common model for social networks. Given an n-node symmetric network generated from a K-community MMSBM, we would like to test $K=1$ versus $K>1$. We first study the degree-based ${\mathit{\chi }}^2$ test and the orthodox Signed Quadrilateral (oSQ) test. These two statistics estimate an order-2 polynomial and an order-4 polynomial of a “signal” matrix, respectively. We derive the asymptotic null distribution and power for both tests. However, for each test, there exists a parameter regime where its power is unsatisfactory. It motivates us to propose a power enhancement (PE) test to combine the strengths of both tests. We show that the PE test has a tractable null distribution and improves the power of both tests. To assess the optimality of PE, we consider a randomized setting, where the n membership vectors are independently drawn from a distribution on the standard simplex. We show that the success of global testing is governed by a quantity ${\mathit{\beta }}_n(K,P,h)$, which depends on the community structure matrix P and the mean vector h of memberships. For each given $(K,P,h)$, a test is called optimal if it distinguishes two hypotheses when ${\mathit{\beta }}_n(K,P,h)\to \mathrm{\infty }$. A test is called optimally adaptive if it is optimal for all $(K,P,h)$. We show that the PE test is optimally adaptive, while many existing tests are only optimal for some particular $(K,P,h)$, hence, not optimally adaptive.

We consider a regression model with errors that are a.s. negative. Thus the regression function is not the expected value of the observations but the right endpoint of their support. We develop two goodness-of-fit tests for the hypotheses that the regression function is an affine function, study the asymptotic distributions of the test statistics in order to approximately fix the sizes of the tests, derive their finite-sample properties based on simulations and apply them to life expectancy data.

We consider a class of parabolic stochastic PDEs on bounded domains $D\subseteq {\mathbb{R}}^d$ that includes the stochastic heat equation but with a fractional power γ of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces $H_r$, with $r<\mathit{\gamma }-d∕2$, we study its power variations in $H_r$ along regular partitions of the time-axis. As the mesh size tends to zero, we find a phase transition at $r=-d∕2$: the solutions have a nontrivial quadratic variation when $r<-d∕2$ and a nontrivial pth order variation for $p=2\mathit{\gamma }∕(\mathit{\gamma }-d∕2-r)>2$ when $r>-d∕2$. More generally, normalized power variations of any order satisfy a genuine law of large numbers in the first case and a degenerate limit theorem in the second case. When $r<-d∕2$, the quadratic variation is given explicitly via an expression that involves the spectral zeta function, which reduces to the Riemann zeta function when $d=1$ and D is an interval.

We derive normal approximation bounds in the Kolmogorov distance for random variables possessing decompositions of Barbour, Karoński, and Ruciński (J. Combin. Theory Ser. B 47 (1989) 125–145). We highlight the example of standardized subgraph counts in the Erdős–Rényi random graph. We prove a bound by generalizing the argumentation of Röllin (Probab. Engrg. Inform. Sci. (2022) 747–773), who used the Stein–Tikhomirov method to prove a bound in the special case of standardized triangle counts. Our bounds match the best available Wasserstein bounds.

Lasso and Dantzig selector are standard procedures able to perform variable selection and estimation simultaneously. This paper is concerned with extending these procedures to spatial point process intensity estimation. We propose adaptive versions of these procedures, develop efficient computational methodologies and derive asymptotic results for a large class of spatial point processes under an original setting where the number of parameters, i.e. the number of spatial covariates considered, increases with the expected number of data points. Both procedures are compared theoretically, in a simulation study, and in a real data example.

In this article, we study the forward integral, in the Russo and Vallois sense, with respect to Hölder continuous stochastic processes Y with exponent bigger than $1∕2$. Here, the integrands have the form $f(Y)$, where f is a bounded variation function. As a consequence of our results, we show that this integral agrees with the generalized Stieltjes integral given by Zähle and that, in the case that Y is fractional Brownian motion, this forward integral is equal to the divergence operator plus a trace term, which is related to the local time of Y. Moreover, the definition of the forward integral allows us to obtain a representation of the solutions to forward stochastic differential equations with a possibly discontinuous coefficient and, as a consequence of our analysis, to figure out some explicit solutions.

Consider two random vectors $\tilde{\mathbf{x}}=A\mathbf{z}+{\mathbf{C}}_1^{1∕2}\mathbf{x}\in {\mathbb{R}}^p$ and $\tilde{\mathbf{y}}=B\mathbf{z}+{\mathbf{C}}_2^{1∕2}\mathbf{y}\in {\mathbb{R}}^q$, where $\mathbf{x}\in {\mathbb{R}}^p$, $\mathbf{y}\in {\mathbb{R}}^q$ and $\mathbf{z}\in {\mathbb{R}}^r$ are independent random vectors with i.i.d. entries of zero mean and unit variance, ${\mathbf{C}}_1$ and ${\mathbf{C}}_2$ are $p\times p$ and $q\times q$ deterministic population covariance matrices, and A and B are $p\times r$ and $q\times r$ deterministic factor loading matrices. With n independent observations of $\tilde{\mathbf{x}}$ and $\tilde{\mathbf{y}}$, we study the sample canonical correlations between them. Under the sharp fourth moment condition on the entries of x, y and z, we prove the BBP transition for the sample canonical correlation coefficients (CCCs). More precisely, if a population CCC is below a threshold, then the corresponding sample CCC converges to the right edge of the bulk eigenvalue spectrum of the sample canonical correlation matrix and satisfies the famous Tracy-Widom law; if a population CCC is above the threshold, then the corresponding sample CCC converges to an outlier that is detached from the bulk eigenvalue spectrum. We prove our results in full generality, in the sense that they also hold for near-degenerate population CCCs and population CCCs that are close to the threshold.

Let $\mathit{\alpha }\in (0,2)$ and $d\in \mathbb{N}$. We consider the stochastic differential equation (SDE) driven by an α-stable process

$$\mathrm{d}{X}_t=b(X_t)\mathrm{d}t+\mathit{\sigma }(X_{t-})\mathrm{d}{L}_t^{\mathit{\alpha }},X_0=x\in {\mathbb{R}}^d,$$

where $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and $\mathit{\sigma }:{\mathbb{R}}^d\to {\mathbb{R}}^d\otimes {\mathbb{R}}^d$ are locally γ-Hölder continuous with $\mathit{\gamma }\in (0\vee {(1-\mathit{\alpha })}^{+},1]$, and $L_t^{\mathit{\alpha }}$ is a d-dimensional symmetric rotationally invariant α-stable process. Under certain dissipative and non-degenerate assumptions on b and σ, we show the V-uniformly exponential ergodicity for the semigroup $P_t$ associated with $\{X_t(x),t\ge 0\}$. Our proofs are mainly based on the heat kernel estimates recently established in (J. Éc. Polytech. Math. 9 (2022) 537–579) to demonstrate the strong Feller property and irreducibility of $P_t$. Interestingly, when α tends to zero, the diffusion coefficient σ can increase faster than the drift b. As an application, we put forward a new heavy-tailed sampling scheme.

Two major research tasks lie at the heart of high dimensional data analysis: accurate parameter estimation and correct support recovery. The existing literature mostly aims for either the best parameter estimation or the best model selection result, however little has been done to understand the potential interaction between the estimation precision and the selection behavior. In this work, our minimax result shows that an estimator’s performance of type I error control directly links with its $L_2$ estimation error rate, and reveals a trade-off phenomenon between the rate of convergence and the false discovery control: to achieve better accuracy, one risks yielding more false discoveries. In particular, we characterize the false discovery control behavior of rate optimal and rate suboptimal estimators under different sparsity regimes, and discover a rigid dichotomy between these two estimators under near-linear and linear sparsity settings. In addition, this work provides a rigorous explanation to the incompatibility phenomenon between selection consistency and rate minimaxity which has been frequently observed in the high dimensional literature.

In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process G, whose covariance function is given by the sinc function. First, we find the convergence of the RTP towards G in the Wasserstein$-1$ distance, which in turn is a consequence of Donsker Theorem. Then, we use this result to derive the rate of convergence between their respective number of zeros. Since the number of real zeros of the RTP is not a continuous function, we use the Kac-Rice formula to express it as the limit of an integral and, in this way, we approximate it by locally Lipschitz continuous functions.

Friedman’s chi-square test is a non-parametric statistical test for r treatments across n trials to assess the null hypothesis that there is no treatment effect. We use Stein’s method with an exchangeable pair coupling to derive a bound on the distance between the distribution of Friedman’s statistic and its limiting chi-square distribution, measured using smooth test functions. Our bound is of the optimal order $n^{-1}$, and also has an optimal dependence on the parameter r, in that the bound tends to zero if and only if $r∕n\to 0$. From this bound, we deduce a Kolmogorov distance bound that decays to zero under the weaker condition $r^{1∕2}∕n\to 0$.

This work contributes to the limited literature on estimating the diffusivity or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that the solution is measured locally in space and over a finite time interval, we show that the augmented maximum likelihood estimator introduced in (Ann. Appl. Probab. 31 (2021) 1–38) for linear SPDEs remains rate-optimal when applied to a large class of semilinear SPDEs. The obtained abstract results are applied to several important classes of SPDEs, including stochastic reaction-diffusion equations. Moreover, we also study the stochastic Burgers equation, as an example with first order nonlinearity, which is a borderline case of the general results. The optimal statistical results are obtained through a precise control of the spatial regularity of the solution and by using higher order fractional $L^p$-Sobolev type spaces. We conclude with numerical examples that validate the theoretical results.

In this paper we consider multivariate Hawkes processes with baseline conditional intensities and reproduction functions that depend on time. This defines a class of locally stationary processes. We discuss estimation of the time-dependent baseline intensities and reproduction functions based on a localized criterion. Theory on stationary Hawkes processes is extended to develop asymptotic theory for the estimator in the locally stationary model.

Exchangeability—in which the distribution of an infinite sequence is invariant to reorderings of its elements—implies the existence of a simple conditional independence structure that may be leveraged in the design of statistical models and inference procedures. In this work, we study a relaxation of exchangeability in which this invariance need not hold precisely. We introduce the notion of local exchangeability—where swapping data associated with nearby covariates causes a bounded change in the distribution. We prove that locally exchangeable processes correspond to independent observations from an underlying measure-valued stochastic process. Using this main probabilistic result, we show that the local empirical measure of a finite collection of observations provides an approximation of the underlying measure-valued process and Bayesian posterior predictive distributions. The paper concludes with applications of the main theoretical results to a model from Bayesian nonparametrics and covariate-dependent permutation tests.

Variable selection in sparse regression models is an important task as applications ranging from biomedical research to econometrics have shown. Especially for higher dimensional regression problems, for which the regression function as the link between response and covariates cannot be directly detected, the selection of informative variables is challenging. Under these circumstances, the Random Forest method is an helpful tool to predict new outcomes while delivering measures for variable selection. One common approach is the usage of the permutation importance. Due to its intuitive idea and flexible usage, it is important to explore circumstances, for which the permutation importance based on Random Forest correctly indicates informative covariates. Regarding the latter, we deliver theoretical guarantees for the validity of the permutation importance measure under specific assumptions such as the mutual independence of the features and prove its (asymptotic) unbiasedness, while under slightly stricter assumptions, consistency of the permutation importance measure is established. An extensive simulation study supports our findings.

In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with diameter measured in Wasserstein distance $W_p$, $p\ge 1$, and for densities with unknown regularity measured on a Besov scale. As sampling domains, we focus on the d-dimensional torus ${\mathbb{T}}^d$, in which case $1\le p\le 2$, and ${\mathbb{R}}^d$, for which $p=1$. We identify necessary and sufficient conditions for the existence of adaptive confidence sets with diameters of the order of the regularity-dependent $W_p$-minimax estimation rate. Interestingly, it appears that the possibility of such adaptation of the diameter depends on the dimension of the underlying space. In low dimensions, $d\le 4$, adaptation to any regularity is possible. In higher dimensions, adaptation is possible if and only if the underlying regularities belong to some bounded interval, whose width can be chosen to be at least $d∕(d-4)$. This contrasts with the $L_p$-theory where, independently of the dimension, adaptation occurs only if regularities lie in a small fixed-width window. When possible, we explicitly construct confidence regions via the method of risk estimation. These are the first results in a statistical approach to adaptive uncertainty quantification with Wasserstein distances. Our analysis and methods extend to weak losses such as Sobolev norms with negative smoothness indices.

Infinite-activity completely random measures (CRMs) have become important building blocks of complex Bayesian nonparametric models. They have been successfully used in various applications such as clustering, density estimation, latent feature models, survival analysis or network science. Popular infinite-activity CRMs include the (generalised) gamma process and the (stable) beta process. However, except in some specific cases, exact simulation or scalable inference with these models is challenging and finite-dimensional approximations are often considered. In this work, we propose a general and unified framework to derive both series representations and finite-dimensional approximations of CRMs. Our framework can be seen as a generalisation of constructions based on size-biased sampling of Poisson point process (Probab. Theory Related Fields 92 (1992) 21–39). It includes as special cases several known series representations and finite approximations as well as novel ones. In particular, we show that one can get novel series representations for the generalised gamma process and the stable beta process. We show how these constructions can be used to derive novel algorithms for posterior inference, including a generalisation of the slice sampler for normalised CRMs mixture models introduced by (J. Comput. Graph. Statist. 20 (2011) 241–259). We also provide some analysis of the truncation error.

The paper discusses multivariate self- and cross-exciting processes. We define a class of multivariate point processes via their corresponding stochastic intensity processes that are driven by stochastic jumps. Essentially, there is a jump in an intensity process whenever the corresponding point process records an event. An attribute of our modelling class is that not only a jump is recorded at each instance, but also its magnitude. This allows large jumps to influence the intensity to a larger degree than smaller jumps. We give conditions which guarantee that the process is stable, in the sense that it does not explode, and provide a detailed discussion on when particular subclasses, which include both linear and non-linear models, are stable. Finally, we fit our model to financial time series data from the S&P 500 and Nikkei 225 indices respectively. We conclude that a non-linear variant from our modelling class fits the data best. This supports the observation that in times of crises (high intensity) jumps tend to arrive in clusters, whereas there are typically longer times between jumps when the markets are calmer. We moreover observe more variability in jump sizes when the intensity is high, than when it is low.

We introduce a class of non-regular estimators in arbitrary normed spaces which include estimators for a non-negative mean, for the squared mean, as well as for the Hodges and Stein estimators. In these cases, the nonparametric bootstrap is consistent on all but a small subset of the underlying parameter space. Bootstrap remedies, such as the $m\text{-out-of-}n$ bootstrap and the oracle bootstrap, have been proposed to mainly solve the inconsistency of the nonparametric bootstrap under a fixed parameter setting. We study the local asymptotic behavior of the estimators and of their bootstrap distributions by allowing the underlying parameter to approach a fixed value at various rates. Our theoretical results determine the precise limiting behavior of the estimators and of their bootstrap distributions in these problems. Simulation results examining the finite sample local performance of the bootstrap estimators are provided.

We present explicit estimates of right and left tails and exact (up to universal, multiplicative constants) estimates of tails and moments of hitting times of Bessel processes. The latter estimates are obtained from more general estimates of moments and tails established for convolutions of elementary mixtures of exponential distributions.

We study the non-parametric estimation of the value of a linear functional evaluated at an unknown density function f with support on ${\mathbb{R}}_{+}$ based on an i.i.d. sample with multiplicative measurement errors. The proposed plug-in estimation procedure combines the estimation of the Mellin transform of the density f and a regularisation of the inverse of the Mellin transform by a spectral cut-off. The attainable accuracy of the estimator is essentially determined by the decay of the upcoming Mellin transforms and the smoothness of the linear functional which we illustrate by different scenarios. As usual the choice of the cut-off parameter is crucial and we propose its data-driven selection inspired by the work of (Goldenshluger and Lepski Ann. Statist. 39 (2011) 1608–1632). By proving matching lower bounds we show that the plug-in estimator with optimally chosen cut-off parameter attains minimax-optimal rates of convergence over Mellin-Sobolev spaces. Furthermore the rate of convergence of the data-driven estimator features at most a deterioration by a logarithmic factor which is widely considered as an acceptable price for adaptation. In particular, our theory covers point-wise estimation of the density f, its derivative and Laplace transform, its associated survival and cumulative distribution function as well as the point-wise estimation of the mean residual life.

Suppose that the arrival rate and the jump distribution of a compound Poisson process change suddenly at an unknown and unobservable time. We want to detect the change as quickly as possible to take counteractions, e.g., to assure top quality of products in a production system, or to stop credit card fraud in a banking system. If we have no prior information about future disorder time, then we typically assume that the disorder is equally likely to happen any time – or has uniform distribution – over a long but finite time horizon. We solve this so-called compound Poisson disorder problem for the practically important case of unknown, unobserved, but uniformly distributed disorder time. The solution hinges on the complete separation of information flow from the hard time horizon constraint, by describing the former with an autonomous time-homogeneous one-dimensional Markov process in terms of which the detection problem translates into a finite horizon optimal stopping problem. For any given finite horizon, the solution is two-dimensional. For cases where the horizon is large and one is unwilling to set a fixed value for it, we give a one-dimensional approximation. Also, we discuss an extension where the disorder may not happen on the given interval with a positive probability. In this extended model, if no detection decision is made by the end of the horizon, then a second level hypothesis testing problem is solved to determine the local parameters of the observed process.

In this paper, we present a new approach to derive series expansions for some Gaussian processes based on harmonic analysis of their covariance function. In particular, we propose a new simple rate-optimal series expansion for fractional Brownian motion. The convergence of the latter series holds in mean square and uniformly almost surely, with a rate-optimal decay of the remainder of the series. We also develop a general framework of convergent series expansions for certain classes of Gaussian processes with stationarity. Finally, an application to optimal functional quantization is described.

In this paper, we carry out a piecewise constant estimator of the density for privatised data. We establish a non-asymptotic oracle inequality for the Hellinger loss and deduce that our estimator is adaptive and rate optimal over a wide range of Besov classes (up to possible logarithmic factors). We also get better estimation rates when the density is not only in a Besov class but also bounded away from 0. These rates are optimal within possible log factors. This result is in contrast to what happens with the ${\mathbb{L}}^2$ loss where the privatised minimax rates over Besov classes can be improved in some cases by assuming the target bounded from above.

We show local asymptotic normality (LAN) for a statistical model of discretely observed ergodic jump-diffusion processes, where the drift coefficient, diffusion coefficient, and jump structure are parametrized. Under the LAN property, we can discuss the asymptotic efficiency of regular estimators, and the quasi-maximum-likelihood and Bayes-type estimators proposed in Shimizu and Yoshida (Stat. Inference Stoch. Process. 9 (2006) 227–277) and Ogihara and Yoshida (Stat. Inference Stoch. Process. 14 (2011) 189–229) are shown to be asymptotically efficient in this model. Moreover, we can construct asymptotically uniformly most powerful tests for the parameters. Unlike with a model for diffusion processes, Aronson-type estimates of the transition density functions do not hold, which makes it difficult to prove LAN. Therefore, instead of Aronson-type estimates, we employ the idea of Theorem 1 in Jeganathan (Sankhyā Ser. A 44 (1982) 173–212) and use the $L^2$ regularity condition. Moreover, we show that local asymptotic mixed normality of a statistical model is implied from that for a model generated by approximated transition density functions under suitable conditions. Together with density approximation by means of thresholding techniques, the LAN property for the jump-diffusion processes is proved.

Developing valid frequency domain bootstrap procedures for integrated periodogram statistics for multivariate time series is a challenging problem. This is mainly due to the fact that the distribution of such statistics depends on the fourth-order moment structure of the underlying multivariate process in nearly every scenario. Exceptions are some very special cases like nonparametric estimators of the spectral density matrix or Gaussian time series. In contrast to the univariate case, even additional structural assumptions – such as linearity of the multivariate process or a standardization of the statistic of interest – do not solve the problem. This paper proposes a new frequency domain bootstrap procedure for multivariate time series, the multivariate frequency domain hybrid bootstrap (MFHB), for integrated periodogram statistics as well as for functions thereof. Asymptotic validity of the MFHB procedure is established for these statistics and for a class of stationary multivariate processes satisfying rather weak dependence conditions ranging clearly beyond linear processes. The finite sample performance of the MFHB is investigated by means of simulations.

This paper derives several novel properties of conditional quantiles viewed as nonlinear operators. The results are organized in parallel to the usual properties of the expectation operator. We first define a τ-conditional quantile random set, relative to any sigma-algebra, as a set of solutions of an optimization problem. Then, well-known properties of unconditional quantiles, as translation invariance, comonotonicity, and equivariance to monotone transformations, are generalized to the conditional case. Moreover, a simple proof for Jensen’s inequality for conditional quantiles is provided. We also investigate continuity of conditional quantiles as operators with respect to different topologies and obtain a novel Fatou’s lemma for quantiles. Conditions for continuity in ${\mathsf{L}}^p$ and weak continuity are also derived. Then, the differentiability properties of quantiles are addressed. We demonstrate the validity of Leibniz’s rule for conditional quantiles for the cases of monotone, as well as separable functions. Finally, although the law of iterated quantiles does not hold in general, we characterize the maximum set of random variables for which this law holds, and investigate its consequences for the infinite composition of conditional quantiles.

We study random graphs with latent geometric structure, where the probability of each edge depends on the underlying random positions corresponding to the two endpoints. We consider the setting where this conditional probability is a general monotone increasing function of the inner product of two vectors; such a function can naturally be viewed as the cumulative distribution function of some independent random variable. A one-parameter family of random graphs, characterized by the variance of this random variable, that smoothly interpolates between a random dot product graph and an Erdős–Rényi random graph, is investigated. Focusing on the dense regime, we prove phase transitions of detecting geometry in these graphs, in terms of the dimension of the underlying geometric space and the variance parameter: When the dimension is high or the variance is large, the graph is similar to an Erdős–Rényi graph with the same edge density; in other parameter regimes, there is a computationally efficient signed triangle statistic that can distinguish them. The proofs make use of information-theoretic inequalities and concentration of measure phenomena.

In this paper, we investigate the exponential ergodicity in a Wasserstein-type distance for a damping Hamiltonian dynamics with state-dependent and non-local collisions, which indeed is a special case of piecewise deterministic Markov processes that is very popular in numerous modelling situations including stochastic algorithms. The approach adopted in this work is based on a combination of the refined basic coupling and the refined reflection coupling for non-local operators. In a certain sense, the main result developed in the present paper is a continuation of the counterpart in (Stochastic Process. Appl. (2022) 146 114–142) on exponential ergodicity of stochastic Hamiltonian systems with Lévy noises and a complement of (Ann. Inst. Henri Poincaré Probab. Stat. 58 (2022a) 916–944) upon exponential ergodicity for Andersen dynamics with constant jump rate functions.

Let $X_1,\dots ,X_n$ be independent and identically distributed random vectors in ${\mathbb{R}}^d$. Suppose $\mathbb{E}{X}_1=0$, $\mathrm{Cov}(X_1)=I_d$, where $I_d$ is the $d\times d$ identity matrix. Suppose further that there exist positive constants $t_0$ and $c_0$ such that $\mathbb{E}{e}^{t_0|X_1|}\le c_0<\mathrm{\infty }$, where $|\cdot |$ denotes the Euclidean norm. Let $W={\sum }_{i=1}^n{X}_i∕\sqrt{n}$ and let Z be a d-dimensional standard normal random vector. Let Q be a $d\times d$ symmetric positive definite matrix whose largest eigenvalue is 1. We prove that for $0\le x\le \mathit{\epsilon }{n}^{1∕6}$,

$$\left|\frac{\mathbb{P}(|Q^{1∕2}W|>x)}{\mathbb{P}(|Q^{1∕2}Z|>x)}-1\right|\le C\left(\frac{1+x^5}{det(Q^{1∕2})n}+\frac{x^6}{n}\right)\text{for}d\ge 5$$

and

$$\left|\frac{\mathbb{P}(|Q^{1∕2}W|>x)}{\mathbb{P}(|Q^{1∕2}Z|>x)}-1\right|\le C\left(\frac{1+x^3}{det(Q^{1∕2})n^{\frac{d}{d+1}}}+\frac{x^6}{n}\right)\text{for}1\le d\le 4,$$

where ε and C are positive constants depending only on $d,t_0$, and $c_0$. This is a first extension of Cramér-type moderate deviation to the multivariate setting with a faster convergence rate than $1∕\sqrt{n}$. The range of $x=o(n^{1∕6})$ for the relative error to vanish and the dimension requirement $d\ge 5$ for the $1∕n$ rate are both optimal. We prove our result using a new change of measure, a two-term Edgeworth expansion for the changed measure, and cancellation by symmetry for terms of the order $1∕\sqrt{n}$.

This paper proves that, under a monotonicity condition, the invariant probability measure of a McKean–Vlasov process can be approximated by weighted empirical measures of some processes including itself. These processes are described by distribution dependent or empirical measure dependent stochastic differential equations constructed from the equation for the McKean–Vlasov process. Convergence of empirical measures is characterized by upper bound estimates for their Wasserstein distances to the invariant measure. Numerical simulations of the mean-field Ornstein–Uhlenbeck process are implemented to demonstrate the theoretical results.

This article considers parameter estimation for a class of Gaussian random fields on ${[0,1)}^d$ that are observed with measurement error and irregularly spaced design sites. This class comprises Gaussian random fields with suitably smooth mean functions and isotropic powered exponential, Matérn or generalized Wendland covariance functions. Under fixed-domain asymptotics, consistent estimators are proposed for three microergodic parameters, namely the nugget, the smoothness parameter and a parameter related to the coefficient of the principal irregular term of the isotropic covariance function. Upper bounds for the convergence rate of these estimators are established. Simulations are conducted to study the finite sample accuracy of the proposed estimators.

Many commonly used test statistics are based on a norm measuring the evidence against the null hypothesis. To understand how the choice of that norm affects power properties of tests in high dimensions, we study the consistency sets of p-norm based tests in the prototypical framework of sequence models with unrestricted parameter spaces, the null hypothesis being that all observations have zero mean. The consistency set of a test is here defined as the set of all arrays of alternatives the test is consistent against as the dimension of the parameter space diverges. We characterize the consistency sets of p-norm based tests and find, in particular, that the consistency against an array of alternatives cannot be determined solely in terms of the p-norm of the alternative. Our characterization also reveals an unexpected monotonicity result: namely that the consistency set is strictly increasing in $p\in (0,\mathrm{\infty })$, such that tests based on higher p strictly dominate those based on lower p in terms of consistency. This monotonicity property allows us to construct novel tests that dominate, with respect to their consistency behavior, all p-norm based tests without sacrificing asymptotic size.

In modern deep learning, there is a recent and growing literature on the interplay between large-width asymptotic properties of deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed weights, and Gaussian stochastic processes (SPs). Motivated by empirical analyses that show the potential of replacing Gaussian distributions with Stable distributions for the NN’s weights, in this paper we present a rigorous analysis of the large-width asymptotic behaviour of (fully connected) feed-forward deep Stable NNs, i.e. deep NNs with Stable-distributed weights. We show that as the width goes to infinity jointly over the NN’s layers, i.e. the “joint growth” setting, a rescaled deep Stable NN converges weakly to a Stable SP whose distribution is characterized recursively through the NN’s layers. Because of the non-triangular structure of the NN, this is a non-standard asymptotic problem, to which we propose an inductive approach of independent interest. Then, we establish sup-norm convergence rates of the rescaled deep Stable NN to the Stable SP, under the “joint growth” and a “sequential growth” of the width over the NN’s layers. Such a result provides the difference between the “joint growth” and the “sequential growth” settings, showing that the former leads to a slower rate than the latter, depending on the depth of the layer and the number of inputs of the NN. Our work extends some recent results on infinitely wide limits for deep Gaussian NNs to the more general deep Stable NNs, providing the first result on convergence rates in the “joint growth” setting.