In recent work, Azadkia and Chatterjee (Ann. Statist. 49 (2021) 3070–3102) laid out an ingenious approach to defining consistent measures of conditional dependence. Their fully nonparametric approach forms statistics based on ranks and nearest neighbor graphs. The appealing nonparametric consistency of the resulting conditional dependence measure and the associated empirical conditional dependence coefficient has quickly prompted follow-up work that seeks to study its statistical efficiency. In this paper, we take up the framework of conditional randomization tests (CRT) for conditional independence and conduct a power analysis that considers two types of local alternatives, namely, parametric quadratic mean differentiable alternatives and nonparametric Hölder smooth alternatives. Our local power analysis shows that conditional independence tests using the Azadkia–Chatterjee coefficient remain inefficient even when aided with the CRT framework, and serves as motivation to develop variants of the approach; cf. Lin and Han (Biometrika 110 (2023) 283–299). As a byproduct, we resolve a conjecture of Azadkia and Chatterjee by proving central limit theorems for the considered conditional dependence coefficients, with explicit formulas for the asymptotic variances.

Spatially inhomogeneous functions, which may be smooth in some regions and rough in other regions, are modelled naturally in a Bayesian manner using so-called Besov priors which are given by random wavelet expansions with Laplace-distributed coefficients. This paper studies theoretical guarantees for such prior measures – specifically, we examine their frequentist posterior contraction rates in the setting of non-linear inverse problems with Gaussian white noise. Our results are first derived under a general local Lipschitz assumption on the forward map. We then verify the assumption for two non-linear inverse problems arising from elliptic partial differential equations, the Darcy flow model from geophysics as well as a model for the Schrödinger equation appearing in tomography. In the course of the proofs, we also obtain novel concentration inequalities for penalized least squares estimators with $$ wavelet penalty, which have a natural interpretation as maximum a posteriori (MAP) estimators. The true parameter is assumed to belong to some spatially inhomogeneous Besov class $$, with $$ sufficiently large. In a setting with direct observations, we complement these upper bounds with a lower bound on the rate of contraction for arbitrary Gaussian priors. An immediate consequence of our results is that while Laplace priors can achieve minimax-optimal rates over $$-classes, Gaussian priors are limited to a (by a polynomial factor) slower contraction rate. This gives information-theoretical justification for the intuition that Laplace priors are more compatible with $$ regularity structure in the underlying parameter.

In this paper, we consider the problem of online asymptotic variance estimation for particle filtering and smoothing. Current solutions for the particle filter rely on the particle genealogy and are either unstable or hard to tune in practice. We propose to mitigate these limitations by introducing a new estimator of the asymptotic variance based on the so called backward weights. The resulting estimator is weakly consistent and trades computational cost for more stability and reduced variance. We also propose a more computationally efficient estimator inspired by the PaRIS algorithm of (Bernoulli 23 (2017) 1951–1996). As an application, particle smoothing is considered and an estimator of the asymptotic variance of the Forward Filtering Backward Smoothing estimator applied to additive functionals is provided.

For birth-and-death processes, we show that every initial distribution is reproduced from the first hitting time distribution. The reproduction is done by applying to the distribution function a differential operator defined through the eigenfunction of the generator. Using the spectral theory for generalized second-order differential operators, we study asymmetric random walks and binary branching processes.

We consider a branching Markov process in continuous time in which the particles evolve independently as spectrally negative Lévy processes. When the branching mechanism is critical or subcritical, the process will eventually die and we may define its overall maximum, i.e. the maximum location ever reached by a particle. The purpose of this paper is to give asymptotic estimates for the survival function of this maximum. In particular, we show that in the critical case the asymptotics is polynomial when the underlying Lévy process oscillates or drifts towards $$, and is exponential when it drifts towards $$.

We study sufficient conditions for a local asymptotic mixed normality property of statistical models. We accommodate the framework of Jeganathan [Sankhyā Ser. A 44 (1982) 173–212] to a triangular array of variable dimension to, in particular, treat high-frequency observations of stochastic processes. When observations are smooth in the Malliavin sense, with the aid of Malliavin calculus techniques by Gobet [Bernoulli 7 (2001) 899–912], we further give tractable sufficient conditions which do not require Aronson-type estimates of the transition density function. The transition density function is even allowed to have zeros. For an application, we prove the local asymptotic mixed normality property of hypoelliptic diffusion models under high-frequency observations, in both complete and partial observation frameworks. The former and the latter extend previous results for elliptic diffusions and for integrated diffusions, respectively.

This paper proves that the separation convergence toward the uniform distribution abruptly occurs at times around $$ for the (time-accelerated by 2) Brownian motion on the sphere with a high dimension n. The arguments are based on a new and elementary perturbative approach for estimating hitting times in a small noise context. The quantitative estimates thus obtained are applied to the strong stationary times constructed in (Arnaudon, Coulibaly-Pasquier and Miclo (2020)) to deduce the wanted cut-off phenomenon.

We provide a new characterisation of Duquesne and Le Gall’s α-stable tree, $$, as the solution of a recursive distributional equation (RDE) of the form $$, where g is a concatenation operator, $$ a sequence of scaling factors, $$, $$, and $$ are i.i.d. trees independent of ξ. This generalises the characterisation of the Brownian Continuum Random Tree proved by Albenque and Goldschmidt, based on self-similarity observed by Aldous. By relating to previous results on a rather different class of RDE, we explore the present RDE and obtain for a large class of similar RDEs that the fixpoint is unique (up to multiplication by a constant) and attractive.

Most of the popular dependence measures for two random variables X and Y (such as Pearson’s and Spearman’s correlation, Kendall’s τ and Gini’s γ) vanish whenever X and Y are independent. However, neither does a vanishing dependence measure necessarily imply independence, nor does a measure equal to 1 imply that one variable is a measurable function of the other. Yet, both properties are natural properties for a convincing dependence measure. In this paper, we present a general approach to transforming a given dependence measure into a new one which exactly characterizes independence as well as functional dependence. Our approach uses the concept of monotone rearrangements as introduced by Hardy and Littlewood and is applicable to a broad class of measures. In particular, we are able to define a rearranged Spearman’s ρ and a rearranged Kendall’s τ which do attain the value 0 if and only if both variables are independent, and the value 1 if and only if one variable is a measurable function of the other. We also present simple estimators for the rearranged dependence measures, prove their consistency and illustrate their finite sample properties by means of a simulation study and a data example.

We detail an approach to developing Stein’s method for bounding integral metrics on probability measures defined on a Riemannian manifold M. Our approach exploits the relationship between the generator of a diffusion on M having a target invariant measure and its characterising Stein operator. We consider a pair of such diffusions with different starting points, and through analysis of the distance process between the pair, derive Stein factors, which bound the solution to the Stein equation and its derivatives. The Stein factors contain curvature-dependent terms and reduce to those currently available for $$, and moreover imply that the bounds for $$ remain valid when M is a flat manifold.

Extremal clusters of stationary processes with long memory can be quite intricate. For certain stationary infinitely divisible processes with subexponential tails an extremal cluster may consist of a single extreme value distributed over a stable regenerative set. This happens both in the case of power-like tails and in the case of certain lighter tails, e.g. lognormal-like tails, In this paper we show that in the case of semi-exponential tails, a new shape of extremal clusters arises. In this case each stable regenerative set supports a random panoply of varying extremes.

We consider a modified quadratic variation of the Hermite process based on some well-chosen increments of this process. These special increments have the very useful property to be independent and identically distributed up to asymptotically negligible remainders. We prove that this modified quadratic variation satisfies a Central Limit Theorem and we derive its rate of convergence under the Wasserstein distance via Stein-Malliavin calculus. As a consequence, we construct, for the first time in the literature related to Hermite processes, a strongly consistent and asymptotically normal estimator for the Hurst parameter.

For multivariate stationary time series many important properties, such as partial correlation, graphical models and autoregressive representations are encoded in the inverse of its spectral density matrix. This is not true for nonstationary time series, where the pertinent information lies in the inverse infinite dimensional covariance matrix operator associated with the multivariate time series. This necessitates the study of the covariance of a multivariate nonstationary time series and its relationship to its inverse. We show that if the rows/columns of the infinite dimensional covariance matrix decay at a certain rate then the rate (up to a factor) transfers to the rows/columns of the inverse covariance matrix. This is used to obtain a nonstationary autoregressive representation of the time series and a Baxter-type bound between the parameters of the autoregressive infinite representation and the corresponding finite autoregressive projection. The aforementioned results lay the foundation for the subsequent analysis of locally stationary time series. In particular, we show that smoothness properties on the covariance matrix transfer to (i) the inverse covariance (ii) the parameters of the vector autoregressive representation and (iii) the partial covariances. All results are set up in such a way that the constants involved depend only on the eigenvalue of the covariance matrix and can be applied in the high-dimensional settings with non-diverging eigenvalues.

We examine the relation between a stochastic version of the rough integral with the symmetric-Stratonovich integral in the sense of regularization. Under mild regularity conditions in the sense of Malliavin calculus, we establish equality between stochastic rough and symmetric-Stratonovich integrals driven by a class of Gaussian processes. As a by-product, we show that solutions of multi-dimensional rough differential equations driven by a large class of Gaussian rough paths they are actually solutions to Stratonovich stochastic differential equations. We obtain almost sure convergence rates of the first-order Stratonovich scheme to rough integrals in the sense of Gubinelli. In case the time-increment of the Malliavin derivative of the integrands is regular enough, the rates are essentially sharp. The framework applies to a large class of Gaussian processes whose the second-order derivative of the covariance function is a sigma-finite non-positive measure on $$ off diagonal.

Inference of mean structure is an important problem in time series analysis. Various tests have been developed to test for different mean structures, for example, the presence of structural breaks, and parametric mean structures. However, many of them are designed for handling specific mean structures, and may lose power upon violation of such structural assumptions. In this paper, we propose a new mean stationarity test built around the signal variance. The proposed test is based on a super-efficient estimator which could achieve a convergence rate faster than $$. It can detect non-constancy of the mean function under serial dependence. It is shown to have promising power, especially in detecting hardly noticeable oscillating structures. The proposal is further generalized to test for smooth trend structures and relative signal variability.

Multivariate point processes (MPPs) are widely applied to model the occurrences of events, e.g., natural disasters, online message exchanges, financial transactions or neuronal spike trains. In the Hawkes process model, the probability of occurrences of future events depend on the past of the process. This model is particularly popular for modelling interactive phenomena such as disease expansion. In this work we consider the nonlinear multivariate Hawkes model, which allows to account for excitation and inhibition between interacting entities. We provide theoretical guarantees for applying nonparametric Bayesian estimation methods in this context. In particular, we obtain concentration rates of the posterior distribution on the parameters, under mild assumptions on the prior distribution and the model. These results also lead to convergence rates of Bayesian estimators. Another object of interest in event-data modelling is to infer the graph of interaction - or Granger causal graph. In this case, we provide consistency guarantees; in particular, we prove that the posterior distribution is consistent on the graph adjacency matrix of the process, as well as a Bayesian estimator based on an adequate loss function.

This paper investigates pooling strategies for tail index and extreme quantile estimation from heavy-tailed data. To fully exploit the information contained in several samples, we present general weighted pooled Hill estimators of the tail index and weighted pooled Weissman estimators of extreme quantiles calculated through a nonstandard geometric averaging scheme. We develop their large-sample asymptotic theory across a fixed number of samples, covering the general framework of heterogeneous sample sizes with different and asymptotically dependent distributions. Our results include optimal choices of pooling weights based on asymptotic variance and MSE minimization. In the important application of distributed inference, we prove that the variance-optimal distributed estimators are asymptotically equivalent to the benchmark Hill and Weissman estimators based on the unfeasible combination of subsamples, while the AMSE-optimal distributed estimators enjoy a smaller AMSE than the benchmarks in the case of large bias. We consider additional scenarios where the number of subsamples grows with the total sample size and effective subsample sizes can be low. We extend our methodology to handle serial dependence and the presence of covariates. Simulations confirm the statistical inferential theory of our pooled estimators. Two applications to real weather and insurance data are showcased.

We introduce the notion of p*-values (p*-variables), which generalizes p-values (p-variables) in several senses. The new notion has four natural interpretations: operational, probabilistic, Bayesian, and frequentist. A main example of a p*-value is a mid p-value, which arises in the presence of discrete test statistics. A unified stochastic representation for p-values, mid p-values, and p*-values is obtained to illustrate the relationship between the three objects. We study several ways of merging arbitrarily dependent or independent p*-values into one p-value or p*-value. Admissible calibrators of p*-values to and from p-values and e-values are obtained with nice mathematical forms, revealing the role of p*-values as a bridge between p-values and e-values. The notion of p*-values becomes useful in many situations even if one is only interested in p-values, mid p-values, or e-values. In particular, deterministic tests based on p*-values can be applied to improve some classic methods for p-values and e-values.

We design sequential tests for a large class of nonparametric null hypotheses based on elicitable and identifiable functionals. Such functionals are defined in terms of scoring functions and identification functions, which are ideal building blocks for constructing nonnegative supermartingales under the null. This in turn yields sequential tests via Ville’s inequality. Using regret bounds from Online Convex Optimization, we obtain rigorous guarantees on the asymptotic power of the tests for a wide range of alternative hypotheses. Our results allow for bounded and unbounded data distributions, assuming that a sub-ψ tail bound is satisfied.

For a class of non-linear stochastic heat equations driven by α-stable white noises for $$ with Lipschitz coefficients, we prove the existence and pathwise uniqueness of $$-valued càdlàg solution to such an equation for $$ by considering a sequence of approximating stochastic heat equations driven by truncated α-stable white noises obtained by removing the big jumps from the original α-stable white noise. If the α-stable white noise is spectrally one-sided, under additional monotonicity assumption on noise coefficients, we further prove a comparison theorem on the $$-valued càdlàg solutions to such an equation. As a consequence, the non-negativity of the $$-valued càdlàg solution is established for the above stochastic heat equation with non-negative initial function.

We consider a supercritical branching random walk where each particle gives birth to a random number of particles of the next generation, which move on the real line, according to a fixed law. Let $$ be the counting measure which counts the number of particles of n-th generation situated in a given region. Under suitable conditions, we establish a Berry-Esseen bound and a Cramér type moderate deviation expansion for $$ with suitable norming.

For the multivariate linear regression model with unknown covariance, the corrected Akaike information criterion is the minimum variance unbiased estimator of the expected Kullback–Leibler discrepancy. In this study, based on the loss estimation framework, we show its inadmissibility as an estimator of the Kullback–Leibler discrepancy itself, instead of the expected Kullback–Leibler discrepancy. We provide improved estimators of the Kullback–Leibler discrepancy that work well in reduced-rank situations and examine their performance numerically.

We present new classes of positive definite kernels on non-standard spaces that are integrally strictly positive definite or characteristic. In particular, we discuss radial kernels on separable Hilbert spaces, and introduce broad classes of kernels on Banach spaces and on metric spaces of strong negative type. The general results are used to give explicit classes of kernels on separable $$ spaces and on sets of measures.

We propose an empirical likelihood ratio (elr) test for comparing any two supervised learning models, which may be nested, non-nested, overlapping, misspecified, or correctly specified. The test compares the prediction losses of models based on the cross-validation. We determine the asymptotic null and alternative distributions of the elr test for comparing two nonparametric learning models under a general framework of convex loss functions. However, the prediction losses from the cross-validation involve repeatedly fitting the models with one observation left out, which leads to a heavy computational burden. We introduce an easy-to-implement elr test which requires fitting the models only once and shares the same asymptotics as the original one. The proposed tests are applied to compare additive models with varying-coefficient models. Furthermore, a scalable distributed elr test is proposed for testing the importance of a group of variables in possibly misspecified additive models with massive data. Simulations show that the proposed tests work well and have favorable finite-sample performance compared to some existing approaches. The methodology is validated on an empirical application.

The intersections of beta-Voronoi, beta-prime-Voronoi and Gaussian-Voronoi tessellations in $$ with an ℓ-dimensional affine subspaces, $$, are shown to be random tessellations of the same type but with different model parameters. In particular, the intersection of a classical Poisson-Voronoi tessellation with an affine subspace is shown to have the same distribution as a certain beta-Voronoi tessellation. The geometric properties of the typical cell and, more generally, typical k-faces, of the sectional Poisson-Voronoi tessellation are studied in detail. It is proved that in high dimensions, that is as $$, the intersection of the d-dimensional Poison-Voronoi tessellation with an affine subspace of fixed dimension ℓ converges to the ℓ-dimensional Gaussian-Voronoi tessellation.

In this paper we investigate the existence of (minimal) supersolutions to BSDEs with mean-reflection on the Z component. We first prove that classical methods to obtain conditions for the existence of supersolutions to BSDEs cannot be applied for this type of constraints. We show that, contrary to BSDEs with mean-reflections on the Y component, we cannot expect a supersolution with a deterministic increasing process K. Nonetheless, we give conditions for the existence of a supersolution for a stochastic component K and under various constraints. Finally, we turn to the existence of minimal supersolution by formalizing some previous arguments on the time-inconsistency of such problems. We formalize some previous arguments on the time-inconsistency of such problems, proving that a minimal supersolution is necessarily a solution in our framework. We apply the results to a replication problem with consumption-investment strategy under law constraints on the investment strategy. We show that the only strategy that might be optimal is the one with no investment.

Piecewise constant priors are routinely used in the Bayesian Cox proportional hazards model for survival analysis. Despite its popularity, large sample properties of this Bayesian method are not yet well understood. This work provides a unified theory for posterior distributions in this setting, not requiring the priors to be conjugate. We first derive contraction rate results for wide classes of histogram priors on the unknown hazard function and prove asymptotic normality of linear functionals of the posterior hazard in the form of Bernstein–von Mises theorems. Second, using recently developed multiscale techniques, we derive functional limiting results for the cumulative hazard and survival function. Frequentist coverage properties of Bayesian credible sets are investigated: we prove that certain easily computable credible bands for the survival function are optimal frequentist confidence bands. We conduct simulation studies that confirm these predictions, with an excellent behavior particularly in finite samples. Our results suggest that the Bayesian approach can provide an easy solution to obtain both the coefficients estimate and the credible bands for survival function in practice.

We consider some Markov jump processes which model structured populations with interactions via density dependence. We propose a Markov construction involving a distinguished individual (spine) which allows us to describe the random tree and random sample at a given time via a change of probability. This spine construction involves the extension of the type space of individuals to include the state of the population. The jump rates off the spine individual can also be modified. We exploit this approach to study issues concerning population dynamics. For single type populations, we derive the phase diagram of a growth fragmentation model with competition as well as the growth of the size of transient birth and death processes which permit multiple births. We also describe the ancestral lineages of a uniform sample in multitype populations.

In this paper, we study the averaging principle for distribution dependent stochastic differential equations with drift in localized $$ spaces. Using Zvonkin’s transformation and estimates for solutions to Kolmogorov equations, we prove that the solutions of the original system strongly and weakly converge to the solution of the averaged system as the time scale ε goes to zero. Moreover, we obtain rates of the strong and weak convergence that depend on p.

We define a new class of Gaussian processes on compact metric graphs such as street or river networks. The proposed models, the Whittle–Matérn fields, are defined via a fractional stochastic differential equation on the compact metric graph and are a natural extension of Gaussian fields with Matérn covariance functions on Euclidean domains to the non-Euclidean metric graph setting. Existence of the processes, as well as some of their main properties, such as sample path regularity are derived. The model class in particular contains differentiable processes. To the best of our knowledge, this is the first construction of a differentiable Gaussian process on general compact metric graphs. Further, we prove an intrinsic property of these processes: that they do not change upon addition or removal of vertices with degree two. Finally, we obtain Karhunen–Loève expansions of the processes, provide numerical experiments, and compare them to Gaussian processes with isotropic covariance functions.

Recently, Chatterjee (2021) introduced a new rank-based correlation coefficient which can be used to measure the strength of dependence between two random variables. This coefficient has already attracted much attention as it converges to the Dette-Siburg-Stoimenov measure (see Dette et al. (2013)), which equals 0 if and only if the variables are independent and 1 if and only if one variable is a function of the other. Further, Chatterjee’s coefficient is computable in (near) linear time, which makes it appropriate for large-scale applications. In this paper, we expand the theoretical understanding of Chatterjee’s coefficient in two directions: (a) First we consider the problem of testing for independence using Chatterjee’s correlation. We obtain its asymptotic distribution under any changing sequence of alternatives converging to the null hypothesis (of independence). We further obtain a general result that gives exact detection thresholds and limiting power for Chatterjee’s test of independence under natural nonparametric alternatives converging to the null. As applications of this general result, we prove a $$ detection boundary for this test and compute explicitly the limiting local power on the detection boundary for popularly studied alternatives in the literature. (b) We then construct a test for non-trivial levels of dependence using Chatterjee’s coefficient. In contrast to testing for independence, we prove that, in this case, Chatterjee’s coefficient indeed yields a minimax optimal procedure with a $$ detection boundary. Our proof techniques rely on Stein’s method of exchangeable pairs, a non-asymptotic projection result, and information theoretic lower bounds.

Let $$ be n pairs of independent random variables. We assume that, for each $$, the conditional distribution of $$ given $$ belongs to a one-parameter exponential family with parameter $$. The statistical goal is to estimate these conditional distributions. We consider a model selection procedure which works based on a general assumption that each of the model is VC-subgraph. We establish a non-asymptotic risk bound for the resulting estimator with respect to a Hellinger-type distance. By leveraging this result, we extend several findings previously explored in Gaussian regression to the regression in exponential families. Specifically, we address the curse of dimensionality by imposing structural assumptions, such as general additive and multiple index structures, on $$. We also study model selection for ReLU neural networks, and provide a concrete example of how ReLU neural networks can achieve a significantly faster convergence rate than traditional models. When $$ is close to a composition of several Hölder functions, we show that under a suitable parametrization of the exponential family, our estimator achieves the same rate of convergence as in the Gaussian case. Combining with a lower bound, the rate is minimax optimal up to a logarithmic term. Finally, we apply the model selection procedure to address adaptation and variable selection problems in exponential families.