In this work, we consider moments of exponential functionals of Lévy processes on a deterministic horizon. We derive two convolutional identities regarding these moments. The first one relates the complex moments of the exponential functional of a general Lévy process up to a deterministic time to those of the dual Lévy process. The second convolutional identity links the complex moments of the exponential functional of a Lévy process, which is not a compound Poisson process, to those of the exponential functionals of its ascending/descending ladder heights on a random horizon determined by the respective local times. As a consequence, we derive a universal expression for the half-negative moment of the exponential functional of any symmetric Lévy process, which resembles in its universality the passage time of symmetric random walks. The ${(n-1∕2)}^{th}$, $n\ge 0$ moments are also discussed. On the other hand, under extremely mild conditions, we obtain a series expansion for the complex moments (including those with negative real part) of the exponential functionals of subordinators. This significantly extends previous results and offers neat expressions for the negative real moments. In a special case, it turns out that the Riemann zeta function is the minus first moment of the exponential functional of the Gamma subordinator indexed in time.

We present a study of a kernel-based two-sample test statistic related to the Maximum Mean Discrepancy (MMD) in the manifold data setting, assuming that high-dimensional observations are close to a low-dimensional manifold. We characterize the test level and power in relation to the kernel bandwidth, the number of samples, and the intrinsic dimensionality of the manifold. Specifically, when data densities p and q are supported on a d-dimensional sub-manifold $\mathcal{M}$ embedded in an m-dimensional space and are Hölder with order β (up to 2) on $\mathcal{M}$, we prove a guarantee of the test power for finite sample size n that exceeds a threshold depending on d, β, and ${\mathrm{\Delta }}_2$ the squared $L^2$-divergence between p and q on the manifold, and with a properly chosen kernel bandwidth γ. For small density departures, we show that with large n they can be detected by the kernel test when ${\mathrm{\Delta }}_2$ is greater than $n^{-2\mathit{\beta }∕(d+4\mathit{\beta })}$ up to a certain constant and γ scales as $n^{-1∕(d+4\mathit{\beta })}$. The analysis extends to cases where the manifold has a boundary and the data samples contain high-dimensional additive noise. Our results indicate that the kernel two-sample test has no curse-of-dimensionality when the data lie on or near a low-dimensional manifold. We validate our theory and the properties of the kernel test for manifold data through a series of numerical experiments.

We propose a new asymptotic test to assess the stationarity of a time series’ mean that is applicable in the presence of both heteroscedasticity and short-range dependence. Our test statistic is composed of Gini’s mean difference of local sample means. To analyse its asymptotic behaviour, we develop new limit theory for U-statistics of strongly mixing triangular arrays under non-stationarity. Most importantly, we show asymptotic normality of the test statistic under the hypothesis of a constant mean and prove the test’s consistency against a very general class of alternatives, including both smooth and abrupt changes in the mean. We propose estimators for all parameters involved, including an adapted subsampling estimator for the long run variance, and show their consistency. Our procedure is practically evaluated in an extensive simulation study and in a data example considering the number of live births in Germany.

We propose a method to construct a joint statistical model for mixed-domain data to analyze their dependence. Multivariate Gaussian and log-linear models are particular examples of the proposed model. It is shown that the functional equation defining the model has a unique solution under fairly weak conditions. The model is characterized by two orthogonal parameters: the dependence parameter and the marginal parameter. To estimate the dependence parameter, a conditional inference together with a sampling procedure is proposed and is shown to provide a consistent estimator. Illustrative examples of data analyses involving penguins and earthquakes are presented.

This article studies the asymptotic properties of Bayesian or frequentist estimators of a vector of parameters related to structural properties of sequences of graphs. The estimators studied originate from a particular class of graphex model introduced by Caron and Fox (J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017) 1295–1366). The analysis is however performed here under very weak assumptions on the underlying data generating process, which may be different from the model of (J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017) 1295–1366) or from a graphex model. In particular, we consider generic sparse graph models, with unbounded degree, whose degree distribution satisfies some assumptions. We show that one can relate the limit of the estimator of one of the parameters to the sparsity constant of the true graph generating process. When taking a Bayesian approach, we also show that the posterior distribution is asymptotically normal. We discuss situations where classical random graphs models, such as configuration models, satisfy our assumptions.

Linear inverse problems for count data arise in a myriad of settings. The latent counts lie on a fibre that is too large to enumerate in most practical problems, but inference can proceed by sampling the fibre. We examine the mixing properties of hit-and-run samplers in this context. In general convergence can be arbitrarily slow. However, there is a class of linear inverse problems for which rapid mixing for uniform fibre sampling is possible, using Markov sub-bases that are of minimum size but yet provide a sufficiently rich range of sampling directions to avoid the need for zig-zagging walks to ensure connectivity. Focussing on such problems, we study a particular class of bases that enjoy these properties under certain easily checkable conditions on the configuration matrix. We also examine the mixing properties of these bases when employing commonly used Poisson models. Our theoretical results provide practical guidance on optimizing these Markov sub-bases.

Let $P_n$ be a random Bernoulli excursion of length $2n$. We show that the area under $P_n$ and the number of peaks of $P_n$ are asymptotically independent. We also show that these statistics have the correlation coefficient asymptotic to $c∕\sqrt{n}$ for large n, where $c<0$, and explicitly compute the coefficient c.

We consider shrinkage estimation of higher-order Hilbert space-valued Bochner integrals in a non-parametric setting. We propose estimators that shrink the U-statistic estimator of the Bochner integral towards a pre-specified target element in the Hilbert space. Depending on the degeneracy of the kernel of the U-statistic, we construct consistent shrinkage estimators and develop oracle inequalities comparing the risks of the U-statistic estimator and its shrinkage version. Surprisingly, we show that the shrinkage estimator designed by assuming complete degeneracy of the kernel of the U-statistic is a consistent estimator even when the kernel is not completely degenerate. This work subsumes and improves upon Muandet et al. (J. Mach. Learn. Res. 17 (2016) 48) and Zhou, Chen and Huang (J. Multivariate Anal. 169 (2019) 166–178), which only handle mean element and covariance operator estimation in a reproducing kernel Hilbert space. We also specialize our results to normal mean estimation and show that for $d\ge 3$, the proposed estimator strictly improves upon the sample mean in terms of the mean squared error.

Copulas are an important tool to study dependencies for data on the real line (or multivariate extensions of this), referred to as linear data. The analogue of copulas for circular data and data on the torus are circulas. This paper studies kernel estimation of circulas, and discusses important issues such as choice of circular kernels and ‘smoothing’ parameters. This leads to some new insights, and some contrasts with results for linear data. Since a circula is a multivariate cumulative distribution with circular uniform marginals, the paper also contributes to kernel estimation of cumulative distributions for toroidal data.

This paper develops a general asymptotic theory of local polynomial (LP) regression for spatial data observed at irregularly spaced locations in a sampling region $R_n\subset {\mathbb{R}}^d$. We adopt a stochastic sampling design that can generate irregularly spaced sampling sites in a flexible manner including both pure increasing and mixed increasing domain frameworks. We first introduce a nonparametric regression model for spatial data defined on ${\mathbb{R}}^d$ and then establish the asymptotic normality of LP estimators with general order $p\ge 1$. We also propose methods for constructing confidence intervals and establish uniform convergence rates of LP estimators. Our dependence structure conditions on the underlying processes cover a wide class of random fields such as Lévy-driven continuous autoregressive moving average random fields. As an application of our main results, we discuss a two-sample testing problem for mean functions and their partial derivatives.

This paper gives a comprehensive treatment of the convergence rates of penalized spline estimators for simultaneously estimating several leading principal component functions, when the functional data is sparsely observed. The penalized spline estimators are defined as the solution of a penalized empirical risk minimization problem, where the loss function belongs to a general class of loss functions motivated by the matrix Bregman divergence, and the penalty term is the integrated squared derivative. The theory reveals that the asymptotic behavior of penalized spline estimators depends on the interesting interplay between several factors, i.e., the smoothness of the unknown functions, the spline degree, the spline knot number, the penalty order, and the penalty parameter. The theory also classifies the asymptotic behavior into seven scenarios and characterizes whether and how the minimax optimal rates of convergence are achievable in each scenario.

We study optimal Markovian couplings of Markov processes, where the optimality is understood in terms of minimization of concave transport costs between evaluations of the coupled processes at corresponding times. We provide explicit constructions of such optimal couplings for one-dimensional finite-activity Lévy processes (continuous-time random walks) whose jump distributions are unimodal but not necessarily symmetric. Remarkably, the optimal Markovian coupling does not depend on the specific concave transport cost. To this end, we combine McCann’s results on optimal transport and Rogers’ results on random walks with a novel uniformization construction that allows us to characterize all Markovian couplings of finite-activity Lévy processes. In particular, we show that the optimal Markovian coupling for finite-activity Lévy processes with non-symmetric unimodal Lévy measures has to allow for non-simultaneous jumps of the two coupled processes.

We provide a unifying approach to central limit type theorems for empirical optimal transport (OT). The limit distribution is given by a supremum of a centered Gaussian process, and we explicitly characterize when it is centered normal or degenerates to a Dirac measure. Moreover, in contrast to recent contributions to distributional limit laws for empirical OT on Euclidean spaces which require centering around its expectation, the limits obtained here are centered around the population quantity, which is well-suited for statistical applications such as goodness-of-fit testing and randomized OT computation. Overall, our distributional limits are valid if one of the population probability measures is of intrinsic dimension at most three. At the heart of our theory lies the Kantorovich duality which represents the OT cost as a supremum over a function class ${\mathcal{F}}_c$ for an underlying sufficiently regular and possibly unbounded cost function c. In this regard, OT is considered as a functional defined on ${\ell }^{\mathrm{\infty }}\left({\mathcal{F}}_c\right)$, the Banach space of bounded functionals from ${\mathcal{F}}_c$ to $\mathbb{R}$ and equipped with uniform norm. We prove the OT functional to be Hadamard directionally differentiable and conclude distributional convergence for increasing sample size via a functional delta method that necessitates weak convergence of an underlying empirical process in ${\ell }^{\mathrm{\infty }}\left({\mathcal{F}}_c\right)$. The latter can be dealt with empirical process theory and requires ${\mathcal{F}}_c$ to be a Donsker class. We give sufficient conditions depending on the dimension of the ground space, the underlying cost function and the probability measures under consideration to guarantee the Donsker property. Altogether, our approach reveals a noteworthy trade-off inherent in central limit theorems for empirical OT: Kantorovich duality requires ${\mathcal{F}}_c$ to be sufficiently rich, while weak convergence of the underlying empirical processes only occurs if ${\mathcal{F}}_c$ is not too complex.

For a continuous-time (integer-time) stochastic process, its distribution at arbitrary time t $(n)$ is often a difficult computation. To use a saddlepoint approximation, its time-indexed moment generating function (MGF) is needed and seldomly is that available. What is often readily available, however, is the Laplace transform (generating function) in time t $(n)$ of this time-indexed MGF which we call a double transform. Such double transforms often take a simple analytic form and we show how they may be inverted to determine the survival function for the process at time t or n. Two general approaches are considered. First, we show that the double-saddlepoint methods initiated by Skovgaard (J. Appl. Probab. 24 (1987) 875–887) may be used by treating the time variable t or n as a random variable with an improper distribution. The second method inverts the double transform in two stages. First, it uses a residue expansion (Butler (J. Appl. Probab. 56 (2019) 307–338; Stoch. Models 39 (2023) 469–501)) to invert it in t or n which is then followed by a single-saddlepoint approximation of the Lugannani-Rice (Adv. in Appl. Probab. 12 (1980) 475–490) type. Applications from renewal theory and renewal reward (cumulative) processes illustrate the remarkable accuracy that results from both of these saddlepoint approximation methods.

We develop a unified approach to bounding the largest and smallest singular values of an inhomogeneous random rectangular matrix, based on the non-backtracking operator and the Ihara-Bass formula for general random Hermitian matrices with a bipartite block structure. We obtain probabilistic upper (respectively, lower) bounds for the largest (respectively, smallest) singular values of a large rectangular random matrix X. These bounds are given in terms of the maximal and minimal ${\ell }_2$-norms of the rows and columns of the variance profile of X. The proofs involve finding probabilistic upper bounds on the spectral radius of an associated non-backtracking matrix B. The two-sided bounds can be applied to the centered adjacency matrix of sparse inhomogeneous Erdős-Rényi bipartite graphs for a wide range of sparsity, down to criticality. In particular, for Erdős-Rényi bipartite graphs $\mathcal{G}(n,m,p)$ with $p=\mathit{\omega }(logn)∕n$, and $m∕n\to y\in (0,1)$, our sharp bounds imply that there are no outliers outside the support of the Marčenko-Pastur law almost surely. This result extends the Bai-Yin theorem to sparse rectangular random matrices.

We study the usual stochastic order between probability measures on preordered topological abelian groups, focusing on asymptotic and catalytic versions thereof. In the asymptotic version, μ dominates ν if the i.i.d. random walk generated by μ first-order dominates the one generated by ν at late times. In the catalytic version, μ dominates ν if there is a third measure τ such that the convolution $\mathrm{\mu }\ast \mathit{\tau }$ first-order dominates $\mathrm{\nu }\ast \mathit{\tau }$. Provided that the preorder on G is induced by a suitably large positive cone and that both measures are compactly supported Radon, our main result gives a sufficient condition for asymptotic and catalytic dominance to hold in terms of a family of inequalities closely related to the cumulant-generating functions. While this sufficient condition requires these inequalities to be strict, the non-strict versions of these inequalities are necessary. This result has been known for $G=\mathbb{R}$, but is new already for ${\mathbb{R}}^n$ with $n>1$. It is a direct application of a recently proven theorem of real algebra, namely a Vergleichsstellensatz for preordered semirings. We finally use our result to derive a formula for the rate at which the probabilities of a random walk decay relative to those of another, now for walks on any preordered topological vector space with compactly supported Radon steps. Taking one of these walks to be deterministic reproduces a version of Cramér’s large deviation theorem for infinite dimensions.

The mathematical forces at work behind Generative Adversarial Networks raise challenging theoretical issues. Motivated by the important question of characterizing the geometrical properties of the generated distributions, we provide a thorough analysis of Wasserstein GANs (WGANs) in both the finite sample and asymptotic regimes. We study the specific case where the latent space is univariate and derive results valid regardless of the dimension of the output space. We show in particular that for a fixed sample size, the optimal WGANs are closely linked with connected paths minimizing the sum of the squared Euclidean distances between the sample points. We also highlight the fact that WGANs are able to approach (for the 1-Wasserstein distance) the target distribution as the sample size tends to infinity, at a given convergence rate and provided the family of generative Lipschitz functions grows appropriately. We derive in passing new results on optimal transport theory in the semi-discrete setting.

We investigate the approximation of high-dimensional target measures as low-dimensional updates of a dominating reference measure. This approximation class replaces the associated density with the composition of: (i) a feature map that identifies the leading principal components or features of the target measure, relative to the reference, and (ii) a low-dimensional profile function. When the reference measure satisfies a subspace ϕ-Sobolev inequality, we construct a computationally tractable approximation that yields certifiable error guarantees with respect to the Amari α-divergences. Our construction proceeds in two stages. First, for any feature map and any α-divergence, we obtain an analytical expression for the optimal profile function. Second, for linear feature maps, the principal features are obtained from eigenvectors of a matrix involving gradients of the log-density. Neither step requires explicit access to normalizing constants. Notably, by leveraging the ϕ-Sobolev inequalities, we demonstrate that these features universally certify approximation errors across the range of α-divergences $\mathit{\alpha }\in (0,1]$. We then propose an application to Bayesian inverse problems and provide an analogous construction with approximation guarantees that hold in expectation over the data. We conclude with an extension of the proposed dimension reduction strategy to nonlinear feature maps.

De Finetti’s theorem, also called the de Finetti–Hewitt–Savage theorem, is a foundational result in probability and statistics. Roughly, it says that an infinite sequence of exchangeable random variables can always be written as a mixture of independent and identically distributed (i.i.d.) sequences of random variables. In this paper, we consider a weighted generalization of exchangeability that allows for weight functions to modify the individual distributions of the random variables along the sequence, provided that – modulo these weight functions – there is still some common exchangeable base measure. We study conditions under which a de Finetti-type representation exists for weighted exchangeable sequences, as a mixture of distributions which satisfy a weighted form of the i.i.d. property. Our approach establishes a nested family of conditions that lead to weighted extensions of other well-known related results as well, in particular, extensions of the zero-one law and the law of large numbers.

We derive normal approximation results for a class of stabilizing functionals of binomial or Poisson point process, that are not necessarily expressible as sums of certain score functions. Our approach is based on a flexible notion of the add-one cost operator, which helps one to deal with the second-order cost operator via suitably appropriate first-order operators. We combine this flexible notion with the theory of strong stabilization to establish our results. We illustrate the applicability of our results by establishing normal approximation results for certain geometric and topological statistics arising frequently in practice. Several existing results also emerge as special cases of our approach.

In this paper, we propose differentially private algorithms for robust (multivariate) mean estimation and inference under heavy-tailed distributions, with a focus on Gaussian differential privacy. First, we provide a comprehensive analysis of the Huber mean estimator with increasing dimensions, including non-asymptotic deviation bound, Bahadur representation, and (uniform) Gaussian approximations. Secondly, we privatize the Huber mean estimator via noisy gradient descent, which is proven to achieve near-optimal statistical guarantees. The key is to characterize quantitatively the trade-off between statistical accuracy, degree of robustness and privacy level, governed by a carefully chosen robustification parameter. Finally, we construct private confidence intervals for the proposed estimator by incorporating a private and robust covariance estimator. Our findings are demonstrated by simulation studies.

We study SDE

$$d{X}_t=b(X_t)dt+A(X_{t-})d{Z}_t,X_0=x\in {ℝ}^d,t\ge 0,$$

where $Z={(Z^1,\dots ,Z^d)}^T$, with $Z^i,i=1,\dots ,d$ being independent one-dimensional symmetric jump Lévy processes, not necessarily identically distributed. In particular, we cover the case when each $Z^i$ is one-dimensional symmetric ${\mathit{\alpha }}_i$-stable process, where ${\mathit{\alpha }}_i\in (0,2)$ are not necessarily equal but satisfy certain balance condition which prevents hypoelliptic effects. Under certain assumptions on b, A and Z we show that the weak solution to the SDE is uniquely defined and is a Markov process. We also provide a representation of the transition probability density and establish Hölder regularity of the corresponding transition semigroup. The method we propose is based on a reduction of an SDE with a drift term to another SDE without such a term but with coefficients depending on time variable. Such a method has the same spirit as the classic characteristic method and seems to be of independent interest.

In this paper, long-time and high-order moment asymptotics for super-Brownian motions (sBm’s) are studied. By using a moment formula for sBm’s (e.g. (Ann. Appl. Probab. 33 (2023) 3872–3915, Theorem 3.1)), precise upper and lower bounds for all positive integer moments at any time $t>0$ of sBm’s for certain initial conditions are achieved. Then, the moment asymptotics as time goes to infinity or as the moment order goes to infinity follow immediately. Additionally, as an application of the two-sided moment bounds, the tail probability estimates of sBm’s are obtained.

We establish an $L_1$-bound between the coefficients of the optimal causal filter applied to the data-generating process and its finite sample approximation. Here, we assume that the data-generating process is a second-order stationary time series with either short or long memory autocovariances. To derive the $L_1$-bound, we first provide an exact expression for the coefficients of the causal filter and their approximations in terms of the absolute convergent series of the multistep ahead infinite and finite predictor coefficients, respectively. Then, we prove a so-called uniform Baxter’s inequality to obtain a bound for the difference between the infinite and finite multistep ahead predictor coefficients in both short and long memory time series. The $L_1$-approximation error bound for the causal filter coefficients can be used to evaluate the performance of the linear predictions of time series through the mean squared error criterion.

In this paper, a new self-normalized and scale invariant statistic $T_n$, which is based on distance correlations, is developed for testing mutual independence of a high-dimensional random vector. The asymptotic normality of the statistic is established under mild moment conditions by assuming both the dimension p of the vector and the sample size n grow to infinity. In particular, the test procedure has the consistency against sparse alternatives where the dependence can be nonlinear and non-monotonic. Technically, the asymptotic normality of the test statistic is built upon martingale decomposition and novel moment method with appropriate combinatorics.

We begin by introducing a class of conditional density estimators based on local polynomial techniques. The estimators are boundary adaptive and easy to implement. We then study the (pointwise and) uniform statistical properties of the estimators, offering characterizations of both probability concentration and distributional approximation. In particular, we establish uniform convergence rates in probability and valid Gaussian distributional approximations for the Studentized t-statistic process. We also discuss implementation issues such as consistent estimation of the covariance function for the Gaussian approximation, optimal integrated mean squared error bandwidth selection, and valid robust bias-corrected inference. We illustrate the applicability of our results by constructing valid confidence bands and hypothesis tests for both parametric specification and shape constraints, explicitly characterizing their approximation errors. A companion $\mathtt{R}$ software package implementing our main results is provided.

This paper derives normal approximation results for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Poisson point process. By combinatorial arguments we express the cumulants of general subgraph counts using sums over connected partition diagrams, after cancellation of terms obtained by Möbius inversion. Using the Statulevičius condition, we deduce convergence rates in the Kolmogorov distance by studying the growth of subgraph count cumulants as the intensity of the underlying Poisson point process tends to infinity. Our analysis covers general subgraphs in the dilute and full random graph regimes, and tree-like subgraphs in the sparse random graph regime.

The validity of statistical inference for panel count data with time-varying covariates depends on the correct specification of within-subject correlation structures; misspecification often leads to questionable inference. To alleviate, robust inference has been proposed for mean models, which implicitly assume monotone mean functions. When covariate values fluctuate with time, however, the assumed monotonicity becomes unrealistic. In this research, we propose a robust inference based on rate models that are free of such constraints. Since the asymptotic variance has no closed form under the rate model, we further develop computationally efficient robust variance estimators using the Expectation-Maximization (EM) algorithm, thus sidestepping the need for computationally intensive numerical methods, which could undermine the robustness. Rigorous theoretical development is provided in support of parameter estimation and inference. Extensive simulation studies demonstrate the superiority of the proposed method. We present a real clinical application to illustrate the use of the proposed method.

We establish nonuniform Berry-Esseen (B-E) bounds for Studentized U-statistics of the rate $1∕\sqrt{n}$ under a third-moment assumption, which covers the t-statistic that corresponds to a kernel of degree 1 as a special case. While an interesting data example raised by Novak (Theory Probab. Appl. 49 (2005) 336–344) can show that the form of the nonuniform bound for standardized U-statistics is actually invalid for their Studentized counterparts, our main results suggest that, the validity of such a bound can be restored by minimally augmenting it with an additive correction term that decays exponentially in n. To our best knowledge, this is the first time that valid nonuniform B-E bounds for Studentized U-statistics have appeared in the literature.

As two crucial tools characterizing regularity properties of stochastic systems, the log-Harnack inequality and Bismut formula have been intensively studied for distribution dependent (McKean-Vlasov) SDEs. However, due to technical difficulties, existing results mainly focus on the case with distribution free noise. In this paper, we introduce a noise decomposition argument to establish the log-Harnack inequality and Bismut formula for SDEs with distribution dependent noise, in both non-degenerate and degenerate situations. As an application, the exponential ergodicity in entropy is investigated.

In this short note, we show that the critical threshold for the percolation of metric graph loop soup on a large class of transient metric graphs (including quasi-transitive graphs such as ${\mathbb{Z}}^d$, $d\ge 3$) is $1∕2$.

Spatially distributed functional data are prevalent in many statistical applications such as meteorology, energy forecasting, census data, disease mapping, and neurological studies. Given their complex and high-dimensional nature, functional data often require dimension reduction methods to extract meaningful information. Inverse regression is one such approach that has become very popular in the past two decades. We study the inverse regression in the framework of functional data observed at irregularly positioned spatial sites. The functional predictor is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect, while the relation between the scalar response and the functional predictor is modeled using the inverse regression framework. For estimation, we consider local linear smoothing with a general weighting scheme, which includes as special cases the schemes under which equal weights are assigned to each observation or to each subject. This framework enables us to present the asymptotic results for different types of sampling plans over time such as non-dense, dense, and ultra-dense. We discuss the domain-expanding infill (DEI) framework for spatial asymptotics, which is a mix of the traditional expanding domain and infill frameworks. The DEI framework overcomes the limitations of traditional spatial asymptotics in the existing literature. Under this unified framework, we develop asymptotic theory and identify conditions that are necessary for the estimated eigen-directions to achieve optimal rates of convergence. Our asymptotic results include pointwise and $L_2$ convergence rates. Simulation studies using synthetic data and an application to a real-world dataset confirm the effectiveness of our methods.

We establish the Bahadur representation of sample quantiles for stabilizing score functionals in stochastic geometry and study local fluctuations of the corresponding empirical distribution function. The scores are obtained from a Poisson process. We apply the results to trimmed and Winsorized means of the score functionals and establish a law of the iterated logarithm for the sample quantiles of the scores.