This paper studies the multi-task high-dimensional linear regression models where the noise among different tasks is correlated, in the moderately high dimensional regime where sample size n and dimension p are of the same order. Our goal is to estimate the covariance matrix of the noise random vectors, or equivalently the correlation of the noise variables on any pair of two tasks. Treating the regression coefficients as a nuisance parameter, we leverage the multi-task elastic-net and multi-task lasso estimators to estimate the nuisance. By precisely understanding the bias of the squared residual matrix and by correcting this bias, we develop a novel estimator of the noise covariance that converges in Frobenius norm at the rate $n^{-1∕2}$ when the covariates are Gaussian distributed with a known covariance matrix. This novel estimator is efficiently computable. Under suitable conditions, the proposed estimator of the noise covariance attains the same rate of convergence as the “oracle” estimator that knows in advance the regression coefficients of the multi-task model. The Frobenius error bounds obtained in this paper also illustrate the advantage of this new estimator compared to a method-of-moments estimator that does not attempt to estimate the nuisance. As byproducts of our techniques, we obtain estimates of the generalization error and out-of-sample error of the multi-task elastic-net and multi-task lasso estimators. Extensive simulation studies are carried out to illustrate the numerical performance of the proposed method.

In this paper, we introduce the concept of isotropic Hilbert-valued spherical random field, thus extending the notion of isotropic spherical random field to an infinite-dimensional setting. We then establish a spectral representation theorem and a functional Schoenberg’s theorem. Following some key results established for the real-valued case, we prove consistency and quantitative central limit theorem for the sample power spectrum operators in the high-frequency regime.

In this paper, we study sparse signal detection problems in “degree corrected” Exponential Random Graph Models (ERGMs). We study the performance of two tests based on conditionally centered sum of degrees/maximum of degrees, for a wide class of such ERGMs. The performance of these tests match the performance of corresponding uncentered tests in the β model (Ann. Statist. 46 (2018) 1288–1317). Focusing on the degree corrected two star ERGM, we show that improved detection is possible at “criticality” using a test based on (unconditional) sum of degrees. In this setting we provide matching lower bounds in all parameter regimes, which is based on correlations estimates between degrees under the alternative, and is of possible independent interest.

In this article, we consider the stochastic fractional diffusion equation

$$\left({\partial }^{\mathit{\beta }}+\frac{\mathrm{\nu }}{2}{\left(-\mathrm{\Delta }\right)}^{\mathit{\alpha }∕2}\right)u(t,x)=\mathit{\lambda }{I}_{0_{+}}^{\mathit{\gamma }}\left[u(t,x)\dot{W}(t,x)\right],t>0,x\in \mathbb{R},$$

where $\mathit{\alpha }>0$, $\mathit{\beta }\in (0,2]$, $\mathit{\gamma }\ge 0$, $\mathit{\lambda }\ne 0$, $\mathrm{\nu }>0$, and $\dot{W}$ is a Gaussian noise which is white or fractional in time and rough in space. We prove the existence and uniqueness of the solution in the Itô-Skorohod sense and obtain the lower and upper bounds for the p-th moment. The Hölder regularity of the solution is also studied.

This article considers the functional linear quantile regression which models the conditional quantile of a scalar response given a functional predictor over a two-dimensional domain. We propose an estimator for the slope function by minimizing the penalized empirical check loss function. Under the framework of reproducing kernel Hilbert space, the minimax rate of convergence for the regularized estimator is established. Using the theory of interpolation spaces on a two- or multi-dimensional domain, we develop a novel result on simultaneous diagonalization of the reproducing and covariance kernels, revealing the interaction of the two kernels in determining the optimal convergence rate of the estimator. Sufficient conditions are provided to show that our analysis applies to many situations, for example, when the covariance kernel is from the Matérn class, and the slope function belongs to a Sobolev space. We implement the interior point method to compute the regularized estimator and illustrate the proposed method by applying it to the hippocampus surface data in the ADNI study.

Some classical mass transportation problems are investigated in a finitely additive setting. Let $\mathrm{\Omega }={\prod }_{i=1}^n{\mathrm{\Omega }}_i$ and $\mathcal{A}={\otimes }_{i=1}^n{\mathcal{A}}_i$, where $({\mathrm{\Omega }}_i,{\mathcal{A}}_i,{\mathrm{\mu }}_i)$ is a (σ-additive) probability space for $i=1,\dots ,n$. Let $c:\mathrm{\Omega }\to [0,\mathrm{\infty }]$ be an $\mathcal{A}$-measurable cost function. Let M be the collection of finitely additive probabilities on $\mathcal{A}$ with marginals ${\mathrm{\mu }}_1,\dots ,{\mathrm{\mu }}_n$. If couplings are meant as elements of M, most classical results of mass transportation theory, including duality and attainability of the Kantorovich inf, are valid without any further assumptions. Special attention is devoted to martingale transport. Let $({\mathrm{\Omega }}_i,{\mathcal{A}}_i)=(\mathbb{R},\mathcal{B}(\mathbb{R}))$ for all i and

$$M_1=\{P\in M:P\ll P^{\ast }\text{and}({\mathrm{\pi }}_1,\dots ,{\mathrm{\pi }}_n)\text{is a}P\text{-martingale}\}$$

where $P^{\ast }$ is a reference probability on $\mathcal{A}$ and ${\mathrm{\pi }}_1,\dots ,{\mathrm{\pi }}_n$ are the canonical projections on $\mathrm{\Omega }={\mathbb{R}}^n$. If $M_1\ne \varnothing$, the Kantorovich inf over $M_1$ is attained, in the sense that $\int cdP={inf}_{Q\in M_1}\int cdQ$ for some $P\in M_1$. Conditions for $M_1\ne \varnothing$ are given as well.

Convergence properties of empirical risk minimizers can be conveniently expressed in terms of the associated population risk. To derive bounds for the performance of the estimator under covariate shift, however, pointwise convergence rates are required. Under weak assumptions on the design distribution, it is shown that least squares estimators (LSE) over 1-Lipschitz functions are also minimax rate optimal with respect to a weighted uniform norm, where the weighting accounts in a natural way for the non-uniformity of the design distribution. This implies that although least squares is a global criterion, the LSE adapts locally to the size of the design density. We develop a new indirect proof technique that establishes the local convergence behavior based on a carefully chosen local perturbation of the LSE. The obtained local rates are then applied to analyze the LSE for transfer learning under covariate shift.

We study representations of a random variable ξ as an integral of an adapted process with respect to the Lebesgue measure. The existence of such representations in two different regularity classes is characterized in terms of the quadratic variation of (local) martingales closed by ξ.

Infinitesimal gradient boosting (Dombry and Duchamps (2021)) is defined as the vanishing-learning-rate limit of the popular tree-based gradient boosting algorithm from machine learning. It is characterized as the solution of a nonlinear ordinary differential equation in an infinite-dimensional function space where the infinitesimal boosting operator driving the dynamics depends on the training sample. We consider the asymptotic behavior of the model in the large sample limit and prove its convergence to a deterministic process. This population limit is again characterized by a differential equation that depends on the population distribution. We explore some properties of this population limit: we prove that the dynamics makes the test error decrease and we consider its long time behavior.

Kernel-weighted test statistics have been widely used in a variety of settings including non-stationary regression, survival analysis, propensity score and panel data models. We develop the limit theory for a kernel-weighted specification test of a parametric conditional mean when the law of the regressors may not be absolutely continuous to the Lebesgue measure and admits non-trivial singular components. In the special case of absolutely continuous measures, our approach weakens the usual regularity conditions. This result is of independent interest and may be useful in other applications that utilize kernel smoothed statistics. Simulations illustrate the non-trivial impact of the distribution of the conditioning variables on the power properties of the test statistic.

Consider the problem of matching two independent i.i.d. samples of size N from two distributions P and Q in ${\mathbb{R}}^d$. For an arbitrary continuous cost function, the optimal assignment problem looks for the matching that minimizes the total cost. We consider instead in this paper the problem where each matching is endowed with a Gibbs probability weight proportional to the exponential of the negative total cost of that matching. Viewing each matching as a joint distribution with N atoms, we then take a convex combination with respect to the above Gibbs probability measure. We show that this resulting random joint distribution converges, as $N\to \mathrm{\infty }$, to the solution of a variational problem, introduced by Föllmer, called the Schrödinger problem. We also prove a limiting Gaussian fluctuation for this convergence in the form of central limit theorems for integrated test functions. This establishes a novel passage for the transition from discrete to continuum in Schrödinger’s lazy gas experiment.

We consider a one-dimensional McKean-Vlasov stochastic differential equation with potential and interaction terms depending on unknown parameters. The sample path is continuously observed on a time interval $[0,2T]$. We assume that the process is in the stationary regime. As this distribution is not explicit, the exact likelihood does not lead to computable estimators. To overcome this difficulty, we consider a kernel estimator of the invariant density based on the sample path on $[0,T]$ and obtain new properties for this estimator. Then, we derive an explicit approximate likelihood using the sample path on $[T,2T]$, including the kernel estimator of the invariant density and study the associated estimators of the unknown parameters. We prove their consistency and asymptotic normality with rate $\sqrt{T}$ as T grows to infinity. Several classes of models illustrate the theory.

Modern neuroimaging research calls for statistical methods that can model dynamic relationships between a functional response and a set of covariates. Current methods, however, remain disparate and limited in their ability to robustly accommodate real-world data and integrate smoothness penalties. In this work, we propose an M-estimation framework for the varying-coefficient model with a functional response that encompasses both mean and quantile regression. To accommodate smoothness regularization and circumvent the stringent conditions on Fourier coefficients or the covariance operator’s eigenvalues imposed by traditional fixed-basis representations, we assume that the functional coefficient resides in a reproducing kernel Hilbert space. We show that our proposed estimator is minimax rate optimal and establish convergence properties of our modified alternating direction method of multipliers algorithm. We further propose combining a weighted M-estimator and a copula model to quantify within-subject spatial dependence to improve estimation accuracy. Simulation studies and a real-world analysis demonstrate the robustness of our proposed methods to outliers.

The imsets of Studený (Probabilistic Conditional Independence Structures (2005) Springer) are an algebraic method for representing conditional independence models. They have many attractive properties when applied to such models, and they are particularly nice for working with directed acyclic graph (DAG) models. In particular, the ‘standard’ imset for a DAG is in one-to-one correspondence with the independences it induces, and hence is a label for its Markov equivalence class. We first present a proposed extension to standard imsets for maximal ancestral graph (MAG) models, using the parameterizing set representation of Hu and Evans (In Proc. 36th Conf. Uncertainty in Artificial Intelligence (2020) PMLR). We show that for many such graphs our proposed imset is perfectly Markovian with respect to the graph, including a class of graphs we refer to as simple MAGs, which includes DAGs as a special case. In these cases the imset provides a scoring criteria by measuring the discrepancy for a list of independences that define the model; this gives an alternative to the usual BIC score that is also consistent, and much easier to compute. We also show that, of independence models that do represent the MAG, the imset we give is minimal. Unfortunately, for some graphs the representation does not represent all the independences in the model, and in certain cases does not represent any at all. For these general MAGs, we refine the reduced ordered local Markov property (Richardson in (Scand. J. Stat. 30 (2003) 145–157)) by a novel graphical tool called power DAGs, and this results in an imset that induces the correct model and which, under a mild condition, can be constructed in polynomial time.

In this paper, we prove a large deviation principle for the empirical measures of a system of weakly interacting diffusions with reflection. We adopt the weak convergence approach. To make this approach work, we show that the sequence of empirical measures of the controlled reflected system will converge to the weak solution of an associated reflected McKean–Vlasov equation.

Stochastic optimal control problems have a long tradition in applied probability, with the questions addressed being of high relevance in a multitude of fields. Even though theoretical solutions are well understood in many scenarios, their practicability suffers from the assumption of known dynamics of the underlying stochastic process, raising the statistical challenge of developing purely data-driven strategies. For the mathematically separated classes of continuous diffusion processes and Lévy processes, we show that developing efficient strategies for related singular stochastic control problems can essentially be reduced to finding rate-optimal estimators with respect to the sup-norm risk of objects associated to the invariant distribution of ergodic processes which determine the theoretical solution of the control problem. From a statistical perspective, we exploit the exponential β-mixing property as the common factor of both scenarios to drive the convergence analysis, indicating that relying on general stability properties of Markov processes is a sufficiently powerful and flexible approach to treat complex applications requiring statistical methods. We show moreover that in the Lévy case—even though per se jump processes are more difficult to handle both in statistics and control theory—a fully data-driven strategy with regret of significantly better order than in the diffusion case can be constructed utilizing spatial ergodicity of a path-time transformation of the Lévy process in form of its overshoots.

We investigate random interlacements on ${\mathbb{Z}}^d$ with $d\ge 3$, and derive the large deviation rate for the probability that the capacity of the interlacement set in a macroscopic box is much smaller than that of the box. As an application, we obtain the large deviation rate for the probability that two independent interlacements have empty intersections in a macroscopic box. Additionally, we prove that conditioning on this event, one of the interlacements will be sparse in terms of capacity within the box. This result is an example of the entropic repulsion phenomenon for random interlacements.

Given a heterogeneous Gaussian sequence model with unknown mean $\mathit{\theta }\in {\mathbb{R}}^d$ and known covariance matrix $\mathrm{\Sigma }=diag({\mathit{\sigma }}_1^2,\dots ,{\mathit{\sigma }}_d^2)$, we study the signal detection problem against sparse alternatives, for known sparsity s. Namely, we characterize how large ${\mathit{ϵ}}^{\ast }>0$ should be, in order to distinguish with high probability the null hypothesis $\mathit{\theta }=0$ from the alternative composed of s-sparse vectors in ${\mathbb{R}}^d$, separated from 0 in $L^t$ norm ($t\in [1,\mathrm{\infty }]$) by at least ${\mathit{ϵ}}^{\ast }$. We find non-asymptotic minimax upper and lower bounds over the minimax separation radius ${\mathit{ϵ}}^{\ast }$ and prove that they are always matching. We also derive the corresponding minimax tests achieving these bounds. Our results reveal new phase transitions regarding the behavior of ${\mathit{ϵ}}^{\ast }$ with respect to the level of sparsity, to the $L^t$ metric, and to the heteroscedasticity profile of Σ. In the case of the Euclidean (i.e. $L^2$) separation, we bridge the remaining gaps in the literature.

The block maxima method is one of the most popular approaches for extreme value analysis with independent and identically distributed observations in the domain of attraction of an extreme value distribution. The lack of a rigorous study on the Bayesian inference in this context has limited its use for statistical analysis of extremes. In this paper we propose an empirical Bayes procedure for inference on the block maxima law and its related quantities. We show that the posterior distributions of the tail index of the data distribution and of the return levels (representative of future extreme episodes) are consistent and asymptotically normal. These properties guarantee the reliability of posterior-based inference. We also establish contraction rates of the posterior predictive distribution, the key tool in Bayesian probabilistic forecasting. Posterior computations are readily obtained via an efficient adaptive Metropolis-Hasting type of algorithm. Simulations show its excellent inferential performances already with modest sample sizes. The utility of our proposal is showcased analysing extreme winds generated by hurricanes in Southeastern US.

We consider Motzkin paths of length L, not fixed at zero at both end points, with constant weights on the edges and general weights on the end points. We investigate, as the length L tends to infinity, the limit behaviors of (a) boundary measures induced by the weights on both end points and (b) the segments of the sampled Motzkin path viewed as a process starting from each of the two end points, referred to as boundary processes. Our first result concerns the case when the induced boundary measures have finite first moments. Our second result concerns when the boundary measure on the right end point is a generalized geometric measure with parameter ${\mathit{\rho }}_1\ge 1$, so that this is an infinite measure and yet it induces a probability measure for random Motzkin path when ${\mathit{\rho }}_1$ is not too large. The two cases under investigation reveal a phase transition. In particular, we show that the limit left boundary processes in the two cases have the same transition probabilities as random walks conditioned to stay non-negative.

We obtain new sharp weighted Poincaré inequalities on Riemannian manifolds for a general class of measures. When specialised to generalised Cauchy measures, this gives a unified and simple proof of the weighted Poincaré inequality for the whole range of parameters, with the optimal spectral gap, the error term and the extremal functions.

The problem of sequential change diagnosis is considered, where observations are obtained on-line, an abrupt change occurs in their distribution, and the goal is to quickly detect the change and accurately identify the post-change distribution, while controlling the false alarm rate. A finite set of alternatives is postulated for the post-change regime, but no prior information is assumed for the unknown change point. A drawback of many algorithms that have been proposed for this problem is the implicit use of pre-change data for determining the post-change distribution. This can lead to very large conditional probabilities of misidentification, given that there was no false alarm, unless the change occurs soon after monitoring begins. A novel, recursive algorithm is proposed and shown to resolve this issue without the use of additional tuning parameters and without sacrificing control of the worst-case delay in Lorden’s sense. A theoretical analysis is conducted for a general family of sequential change diagnosis procedures, which supports the proposed algorithm and revises certain state-of-the-art results. Additionally, a novel, comprehensive method is proposed for the design and evaluation of sequential change diagnosis algorithms. This method is illustrated with simulation studies, where existing procedures are compared to the proposed.

We consider degenerate diffusion equations of the form ${\partial }_t{p}_t=\mathrm{\Delta }f(p_t)$ on a bounded domain and subject to no-flux boundary conditions, for a class of nonlinearities f that includes the porous medium equation. We derive for them a trajectorial analogue of the entropy dissipation identity, which describes the rate of entropy dissipation along every path of the diffusion. In line with the recent work (Theory Probab. Appl. 66 (2022) 668–707), our approach is based on applying stochastic analysis to the underlying probabilistic representations, which in our context are stochastic differential equations with normal reflection on the boundary. This trajectorial approach also leads to a new derivation of the Wasserstein gradient flow property for nonlinear diffusions, as well as to a simple proof of the HWI inequality in the present context.

In many scientific applications the aim is to infer a function which is smooth in some areas, but rough or even discontinuous in other areas of its domain. Such spatially inhomogeneous functions can be modelled in Besov spaces with suitable integrability parameters. In this work we study adaptive Bayesian inference over Besov spaces, in the white noise model from the point of view of rates of contraction, using p-exponential priors, which range between Laplace and Gaussian and possess regularity and scaling hyper-parameters. To achieve adaptation, we employ empirical and hierarchical Bayes approaches for tuning these hyper-parameters. Our results show that, while it is known that Gaussian priors can attain the minimax rate only in Besov spaces of spatially homogeneous functions, Laplace priors lead to adaptive or nearly adaptive procedures in both Besov spaces of spatially homogeneous functions and Besov spaces permitting spatial inhomogeneities.

A Peskun ordering between two samplers, implying a dominance of one over the other, is known among the Markov chain Monte Carlo community for being a remarkably strong result. It is however also known for being a result that is notably difficult to establish. Indeed, one has to prove that the probability to reach a state y from a state x, using a sampler, is greater than or equal to the probability using the other sampler, and this must hold for all pairs $(\mathbf{x},\mathbf{y})$ such that $\mathbf{x}\ne \mathbf{y}$. We provide in this paper a weaker version that does not require an inequality between the probabilities for all these states: essentially, the dominance holds asymptotically, as a varying parameter grows without bound, as long as the states for which the probabilities are greater than or equal to belong to a mass-concentrating set. The weak ordering turns out to be useful to compare lifted samplers for partially-ordered discrete state-spaces with their Metropolis–Hastings counterparts. An analysis in great generality yields a qualitative conclusion: they asymptotically perform better in certain situations (and we are able to identify them), but not necessarily in others (and the reasons why are made clear). A quantitative study in a specific context of graphical-model simulation is also conducted.

This paper proposes convex relaxation based robust methods to recover approximately low-rank matrices in the presence of heavy-tailed and asymmetric errors, allowing for heteroscedasticity. We focus on three archetypal applications in matrix recovery: matrix compressed sensing, matrix completion and multitask regression. Statistically, we provide sub-Gaussian-type deviation bounds when the noise variables only have bounded variances in each aforementioned setting. Improving upon the earlier results in Fan, Wang and Zhu (Ann. Statist. 49 (2021) 1239–1266), the convergence rates of our estimators are proportional to the noise scale under matrix sensing and multitask regression settings, and thus diminish to 0 in the noiseless case. Computationally, we propose a matrix version of the local adaptive majorize-minimization algorithm, which is much faster than the alternating direction method of multiplier used in previous work and is scalable to large datasets. Numerical experiments demonstrate the advantage of our methods over their non-robust counterparts and corroborate the theoretical findings that the convergence rates are proportional to the noise scale.

Let T be a function of Hermite rank one and let ${\{G(t)\}}_{t\in \mathbb{R}}$ be a mean-square continuous stationary Gaussian process with long-range dependence. We explore the finite-dimensional distributions of the second-order scattering transform of the process $X={\{T(G(t))\}}_{t\in \mathbb{R}}$ when all the scale parameters go to infinity simultaneously. For frequently used wavelets, we find a constraint on the ratio of the scale parameters of the wavelet transform within the first and second layers such that the limit exists. The constraint is explicitly expressed in terms of the Hurst index of the long-range dependent inputs and the gap between the indices of the first and second non-zero coefficients in the Hermite expansion of the function T. Under the constraint on the ratio of the scale parameters, we prove that the rescaled second-order scattering transform converges in the finite-dimensional distribution sense to a chi process of degree one. The limiting process is expressed in terms of the Fourier transform of mother wavelet and the Hurst index of long-range dependence.

We consider the problem of detecting jumps in an otherwise smoothly evolving trend whilst the covariance and higher-order structures of the system can experience both smooth and abrupt changes over time. The number of jump points is allowed to diverge to infinity with the jump sizes possibly shrinking to zero. The method is based on a multiscale application of an optimal jump-pass filter to the time series, where the scales are dense between admissible lower and upper bounds. For a wide class of non-stationary time series models and trend functions, the proposed method is shown to be able to detect all jump points within a nearly optimal range with a prescribed probability asymptotically under mild conditions. For a time series of length n, the computational complexity of the proposed method is $O(n)$ for each scale and $O(n{log}^{1+\mathit{ϵ}}n)$ overall, where ϵ is an arbitrarily small positive constant. Numerical studies show that the proposed jump testing and estimation method performs robustly and accurately under complex temporal dynamics.

We prove a law of large numbers and invariance principles for the tagged particle in the asymmetric exclusion process with long jumps when the process starts from its equilibrium measure.

Theory and methodology of nonparametric sharp minimax estimation of the conditional hazard rate function of a right censored lifetime given a continuous covariate are developed. The theory, using an oracle’s approach, shows how the conditional hazard and nuisance functions affect rate and constant of the mean integrated squared error (MISE) convergence. The methodology suggests a data-driven estimator matching performance of the oracle. Further, if the lifetime is independent of the covariate, the estimator recognizes that and the MISE converges with the univariate rate. Then the setting is extended to a vector of continuous and ordinal/nominal categorical predictors, and an estimator performing adaptation to smoothness and dimensionality of conditional hazard is suggested. Practical examples devoted to reducing potent greenhouse gas emissions are presented.

We consider the problem of testing for long-range dependence in time-varying coefficient regression models, where the covariates and errors are locally stationary, allowing complex temporal dynamics and heteroscedasticity. We develop KPSS, R/S, V/S, and K/S-type statistics based on the nonparametric residuals. Under the null hypothesis, the local alternatives as well as the fixed alternatives, we derive the limiting distributions of the test statistics. As the four types of test statistics could degenerate when the time-varying mean, variance, long-run variance of errors, covariates, and the intercept lie in certain hyperplanes, we show the bootstrap-assisted tests are consistent under both degenerate and non-degenerate scenarios. In particular, in the presence of covariates the exact local asymptotic power of the bootstrap-assisted tests can enjoy the same order as that of the classical KPSS test of long memory for strictly stationary series. The asymptotic theory is built on a new Gaussian approximation technique for locally stationary long-memory processes with short-memory covariates, which is of independent interest. The effectiveness of our tests is demonstrated by extensive simulation studies and real data analysis.

We study the large-sample properties of sparse M-estimators in the presence of pseudo-observations. Our framework covers a broad class of semi-parametric copula models, for which the marginal distributions are unknown and replaced by their empirical counterparts. It is well known that the latter modification significantly alters the limiting laws compared to usual M-estimation. We establish the consistency and the asymptotic normality of our sparse penalized M-estimator and we prove the asymptotic oracle property with pseudo-observations, possibly in the case when the number of parameters is diverging. Our framework allows to manage copula-based loss functions that are potentially unbounded. Additionally, we state the weak limit of multivariate rank statistics for an arbitrary dimension and the weak convergence of empirical copula processes indexed by maps. We apply our inference method to Canonical Maximum Likelihood losses with Gaussian copulas, mixtures of copulas or conditional copulas. The theoretical results are illustrated by two numerical experiments.

We derive both Azuma-Hoeffding and Burkholder-type inequalities for partial sums over a rectangular grid of dimension d of a random field satisfying a weak dependency assumption of projective type: the difference between the expectation of an element of the random field and its conditional expectation given the rest of the field at a distance more than δ is bounded, in $L_p$ distance, by a known decreasing function of δ. The analysis is based on the combination of a multi-scale approximation of random sums by martingale difference sequences, and of a careful decomposition of the domain. The obtained results extend previously known bounds under comparable hypotheses, and do not use the assumption of commuting filtrations.

This paper discusses the problem of choosing the optimal block length for two block bootstrap methods designed for periodically correlated processes. These are the Generalized Seasonal Block Bootstrap and the Extension of Moving Block Bootstrap. Two estimation problems are considered: the overall mean and the seasonal means. In both cases, the optimal block length is obtained by minimizing the mean squared error of the corresponding bootstrap variance estimator and in all cases it is proportional to the cube root of the sample size and should be a multiple of the period length plus one observation to avoid some bias. Finally, the results of the performed simulation are presented, in which optimal blocks lengths are estimated for several periodically correlated time series.