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We deal with the problem of optimal estimation of linear functionals constructed from the missed values of a continuous time stochastic process ξ ⁢ (t) with periodically stationary increments at points t ∈ [ 0 ; (N + 1) ⁢ T ] based on observations of this process with periodically stationary noise. To solve the problem, a sequence of stochastic functions { ξ j (d) ⁢ (u) = ξ j (d) ⁢ (u + j ⁢ T , τ) , u ∈ [ 0 , T) , j ∈ ℤ } is constructed. It forms an L 2 ⁢ ([ 0 , T) ; H) -valued stationary increment sequence { ξ j (d) , j ∈ ℤ } or corresponding to it an (infinite-dimensional) vector stationary increment sequence { ξ → j (d) = (ξ k ⁢ j (d) , k = 1 , 2 , ...) ⊤ , j ∈ ℤ } . In the case of a known spectral density, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given. [ABSTRACT FROM AUTHOR]

This paper focuses on nonparametrically estimating the probability $ \mathbb {P} \left (X_t \lt Y_t \right) $ P (X t < Y t) when two strongly mixing stationary processes, $ X_t $ X t and $ Y_t $ Y t , are observed with additional errors at discrete time points $ t_j = j\Delta $ t j = j Δ (where Δ is a positive constant). This problem has practical significance in some applications where data is time-dependent random variable sequences (generated from stochastic processes) like time-dependent stress–strength reliability modelling and receiver operating characteristic (ROC) curve analysis. We extend our analysis to account for errors generated from another strongly mixing stationary processes and derive convergence rate and asymptotic normality results for the estimator under supersmooth error distributions. Through applications and simulations, we illustrate the properties of our estimator. [ABSTRACT FROM AUTHOR]

We consider Palm distributions arising in a Markov process with time homogeneous transitions which is jointly stationary with multiple point processes. Motivated by a BAR approach studied in the recent paper (Braverman et al. in the BAR approach for multi-class queueing networks with SBP service policies, 2023), we are interested in two problems; when this Markov process inherits the same Markov structure under the Palm distributions, and how the state changes at counting instants of the point processes can be handled to derive stationary equations when there are simultaneous counts and each of them influences the state changes. We affirmatively answer the first problem, and propose a framework for resolving the second problem, which is applicable to a general stationary process, which is not needed to be Markov. We also discuss how those results can be applied in deriving BAR's for the diffusion approximation of queueing models in heavy traffic. In particular, as their new application, the heavy traffic limit of the stationary distribution is derived for a single-server queue with a finite waiting room. [ABSTRACT FROM AUTHOR]

We define the co-spectral radius of inclusions ${\mathcal S}\leq {\mathcal R}$ of discrete, probability- measure-preserving equivalence relations as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more general version of the 2–3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius. [ABSTRACT FROM AUTHOR]

When modelling unbounded counts, their marginals are often assumed to follow either Poisson (Poi) or negative binomial (NB) distributions. To test such null hypotheses, we propose goodness-of-fit (GoF) tests based on statistics relying on certain moment properties. By contrast to most approaches proposed in the count-data literature so far, we do not restrict ourselves to specific low-order moments, but consider a flexible class of functions of generalized moments to construct model-diagnostic tests. These cover GoF-tests based on higher-order factorial moments, which are particularly suitable for the Poi- or NB-distribution where simple closed-form expressions for factorial moments of any order exist, but also GoF-tests relying on the respective Stein's identity for the Poi- or NB-distribution. In the time-dependent case, under mild mixing conditions, we derive the asymptotic theory for GoF tests based on higher-order factorial moments for a wide family of stationary processes having Poi- or NB-marginals, respectively. This family also includes a type of NB-autoregressive model, where we provide clarification of some confusion caused in the literature. Additionally, for the case of independent and identically distributed counts, we prove asymptotic normality results for GoF-tests relying on a Stein identity, and we briefly discuss how its statistic might be used to define an omnibus GoF-test. The performance of the tests is investigated with simulations for both asymptotic and bootstrap implementations, also considering various alternative scenarios for power analyses. A data example of daily counts of downloads of a TeX editor is used to illustrate the application of the proposed GoF-tests. [ABSTRACT FROM AUTHOR]

We investigate the problem of joint statistical estimation of several parameters for a stochastic differential equation driven by an additive fractional Brownian motion. Based on discrete-time observations of the model, we construct an estimator of the Hurst parameter, the diffusion parameter and the drift, which lies in a parametrised family of coercive drift coefficients. Our procedure is based on the assumption that the stationary distribution of the SDE and of its increments permits to identify the parameters of the model. Under this assumption, we prove consistency results and derive a rate of convergence for the estimator. Finally, we show that the identifiability assumption is satisfied in the case of a family of fractional Ornstein–Uhlenbeck processes and illustrate our results with some numerical experiments. [ABSTRACT FROM AUTHOR]

This paper introduces stochastic covariance models in Hilbert spaces with stationary affine instantaneous covariance processes. We explore the applications of these models in the context of forward curve dynamics within fixed-income and commodity markets. The affine instantaneous covariance process is defined on positive self-adjoint Hilbert–Schmidt operators, and we prove the existence of a unique limit distribution for subcritical affine processes, provide convergence rates of the transition kernels in the Wasserstein distance of order p ∈ [ 1 , 2 ] , and give explicit formulas for the first two moments of the limit distribution. Our results allow us to introduce affine stochastic covariance models in the stationary covariance regime and to investigate the behaviour of the implied forward volatility for large forward dates in commodity forward markets. [ABSTRACT FROM AUTHOR]

Matrix-valued (multivariate) correlation functions are increasingly used within both the statistics and machine learning communities, but their properties have been studied to a limited extent. The motivation of this paper comes from the fact that the celebrated local stationarity construction for scalar-valued correlations has not been considered for the matrix-valued case. The main reason is a lack of theoretical support for such a construction. We explore the problem of extending a matrix-valued correlation from a d-dimensional ball with arbitrary radius into the d-dimensional Euclidean space. We also consider such a problem over product spaces involving the d-dimensional ball with arbitrary radius. We then provide a useful architecture to matrix-valued local stationarity by defining the class of p-exponentially convex matrix-valued functions, and characterize such a class as scale mixtures of the d-Schoenberg kernels against certain families of measures. We exhibit bijections from such a class into the class of positive semidefinite matrix-valued functions and we extend exponentially convex matrix-valued functions from d-dimensional balls into the d-dimensional Euclidean space. We finally provide similar results for the case of function-valued correlations defined over certain Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this article, we discuss some geometric infinitely divisible (gid) random variables using the Laplace exponents which are Bernstein functions and study their properties. The distributional properties and limiting behavior of the probability densities of these gid random variables at 0 + are studied. The autoregressive (AR) models with gid marginals are introduced. Further, the first order AR process is generalized to kth order AR process. We also provide the parameter estimation method based on conditional least square and method of moments for the introduced AR(1) process. We also apply the introduced AR(1) model with geometric inverse Gaussian marginals on the household energy usage data which provide a good fit as compared to normal AR(1) data. [ABSTRACT FROM AUTHOR]

We give rates of convergence in the almost sure invariance principle for sums of dependent random variables with semi-exponential tails, whose coupling coefficients decrease at a sub-exponential rate. We show that the rates in the strong invariance principle are in powers of log n . We apply our results to iid products of random matrices. [ABSTRACT FROM AUTHOR]

AbstractPatient arrivals at a hospital typically occur in clusters and the arrivals count data exhibits over-dispersion. To model these characteristics, the Lévy-type processes based on Poisson-stopped sum distributions are proposed as alternatives to the non-homogeneous Poisson process (NHPP), a popular model for hospital arrivals. The arrival rate for the NHPP is a deterministic function of time that does not account for arrivals in clusters while the genesis of Poisson-stopped sum distributions arises from modeling event clusters. Data on daily scheduled patient arrivals over 15-minute intervals collected from a Malaysian public hospital were used to illustrate the application of the Levy-type processes in healthcare management. It is shown that the proposed Lévy-type negative binomial and Lévy-type Thomas processes fit the data better than the NHPP. In addition, the Levy-type processes are computationally simpler and hence, it is envisaged that they will be potentially useful for implementation in staff scheduling and resource allocations. [ABSTRACT FROM AUTHOR]

In this paper, the problem of identifying Multiple-Input-Single-Output (MISO) systems with fractional models from noisy input-output available data is studied. The proposed idea is to use Higher-Order-Statistics (HOS), like fourth-order cumulants (foc), instead of noisy measurements. Thus, a fractional fourth-order cumulants based-simplified and refined instrumental variable algorithm (frac-foc-sriv) is first developed. Assuming that all differentiation orders are known a priori, it consists in estimating the linear coefficients of all Single-Input-Single-Output (SISO) sub-models composing the MISO model. Then, the frac-foc-sriv algorithm is combined with a nonlinear optimization technique to estimate all the parameters: coefficients and orders. The performances of the developed algorithms are analyzed using numerical examples. Thanks to fourth-order cumulants, which are insensitive to Gaussian noise, and the iterative strategy of the instrumental variable algorithm, the parameters estimation is consistent. [ABSTRACT FROM AUTHOR]

We prove homogenization for viscous Hamilton-Jacobi equations with a Hamiltonian of the form G (p) + V (x , ω) for a wide class of stationary ergodic random media in one space dimension. The momentum part G(p) of the Hamiltonian is a general (nonconvex) continuous function with superlinear growth at infinity, and the potential V (x , ω) is bounded and Lipschitz continuous. The class of random media we consider is defined by an explicit hill and valley condition on the diffusivity-potential pair which is fulfilled as long as the environment is not "rigid". [ABSTRACT FROM AUTHOR]

Stute (1991) introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: r (k) (φ , t) : = E [ φ (Y 1 , ... , Y k) | (X 1 , ... , X k) = t ] , for t ∈ R d k. This article deals with a quite general non parametric statistical curve estimation setting including the Stute estimator as a particular case. The class of "delta sequence estimators" is defined and treated here. This class includes also the orthogonal series and histogram methods. The theoretical results concerning the exponential inequalities and the asymptotic normality, established in this article, are (or will be) key tools for many further developments in functional estimation. As a by-product of our proofs, we state consistency results for the delta sequences conditional U-statistics estimator, under the random censoring. Potential applications include discrimination problems, metric learning and multipartite ranking, Kendall rank correlation coefficient, generalized U-statistics, and set indexed conditional U-statistics. [ABSTRACT FROM AUTHOR]

This paper addresses the estimation of the second-order structure of a manifold cross-time random field (RF) displaying spatially varying Long Range Dependence (LRD), adopting the functional time series framework introduced in Ruiz-Medina (Fract Calc Appl Anal 25:1426–1458, 2022). Conditions for the asymptotic unbiasedness of the integrated periodogram operator in the Hilbert–Schmidt operator norm are derived beyond structural assumptions. Weak-consistent estimation of the long-memory operator is achieved under a semiparametric functional spectral framework in the Gaussian context. The case where the projected manifold process can display Short Range Dependence (SRD) and LRD at different manifold scales is also analyzed. The performance of both estimation procedures is illustrated in the simulation study, in the context of multifractionally integrated spherical functional autoregressive–moving average (SPHARMA(p,q)) processes. [ABSTRACT FROM AUTHOR]

We study stochastic homogenization for convex integral functionals u ↦ ∫ D W (ω , x ε , ∇ u) d x , where u : D ⊂ R d → R m , defined on Sobolev spaces. Assuming only stochastic integrability of the map ω ↦ W (ω , 0 , ξ) , we prove homogenization results under two different sets of assumptions, namely ∙ 1 W satisfies superlinear growth quantified by the stochastic integrability of the Fenchel conjugate W ∗ (· , 0 , ξ) and a certain monotonicity condition that ensures that the functional does not increase too much by componentwise truncation of u, ∙ 2 W is p-coercive in the sense | ξ | p ≤ W (ω , x , ξ) for some p > d - 1 . Condition ∙ 2 directly improves upon earlier results, where p-coercivity with p > d is assumed and ∙ 1 provides an alternative condition under very weak coercivity assumptions and additional structure conditions on the integrand. We also study the corresponding Euler–Lagrange equations in the setting of Sobolev-Orlicz spaces. In particular, if W (ω , x , ξ) is comparable to W (ω , x , - ξ) in a suitable sense, we show that the homogenized integrand is differentiable. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for the almost sure central limit theorem started at a point, known under the name of quenched central limit theorem. This is achieved by using a new idea of conditioning with respect to both the past and the future of the Markov chain. As applications, we provide a new sufficient projective condition for the quenched CLT. [ABSTRACT FROM AUTHOR]

Vaccination and social media play pivotal roles in affecting disease transmission. Research has shown that disease transmission can be subject to random events. Consequently, we develop a stochastic infectious disease dynamical model that incorporates saturated media coverage and vaccination age. The Itô's formula and the Lyapunov function method are applied to study the extinction behavior of the disease and the existence of a unique ergodic stationary distribution. The findings suggest that the media effect is delayed and cannot eliminate the disease completely. To directly control disease transmission, a combination of high-intensity noise disturbance and low vaccine wane rate is required. Furthermore, the shorter the disease incubation period, the more difficult it is to control. [ABSTRACT FROM AUTHOR]

We consider the class of radial exponentially convex functions defined over n-dimensional balls with finite or infinite radii. We provide characterization theorems for these classes, as well as Rudin's type extension theorems for radial exponentially convex functions defined over n-dimensional balls into radial exponentially convex functions defined over the whole n-dimensional Euclidean space. We furthermore establish inversion theorems for the measures, termed here n-Nussbaum measures, associated with integral representations of radial exponentially convex functions. This in turn allows obtaining recurrence relations between 1-Nussbaum measures and n-Nussbaum measures for a given integer n greater than 1. We also provide a up to now unknown catalogue of radial exponentially convex functions and associated n-Nussbaum measures. We finally turn our attention into componentwise radial exponential convexity over product spaces, with a Rudin extension result and analytical examples of exponentially convex functions and associated Nussbaum measures. As a byproduct, we obtain a parametric model of nonseparable stationary space-time covariance functions that do not belong to the well-known Gneiting class. [ABSTRACT FROM AUTHOR]

The correlation integral of a time series is a normalized coefficient that represents the number of close pairs of points of the series lying in phase space. It has been widely studied in a number of disciplines such as phisycs, mechanical engineering, bioengineering, among others, allowing the estimation of the dimension of an attractor in a chaotic regimen. The computation of the dimension of an attractor allows to distinguish deterministic behavior in stochastic processes with a weak structure on the noise. In this paper, we establish a power law for the limiting expected value of the correlation integral for Gaussian stationary time series. Examples with linear and nonlinear time series are used to illustrate the result. [ABSTRACT FROM AUTHOR]

Analogously to the well-known Langevin Monte Carlo method, in this article we provide a method to sample from a target distribution π by simulating a solution of a stochastic differential equation. Hereby, the stochastic differential equation is driven by a general Lévy process which—unlike the case of Langevin Monte Carlo—allows for non-smooth targets. Our method will be fully explored in the particular setting of target distributions supported on the half-line (0 , ∞) and a compound Poisson driving noise. Several illustrative examples conclude the article. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study a one sided tempered stable first order autoregressive model called TAR(1). Under the assumption of stationarity of the model, the marginal probability density function of the error term is found. It is shown that the distribution of the error term is infinitely divisible. Parameter estimation of the introduced TAR(1) process is done by adopting the conditional least square and method of moments based approach and the performance of the proposed methods are evaluated on simulated data. Also, we study an autoregressive model of order one with tempered stable innovations. Using appropriate test statistic it is shown that the model fits very well on real and simulated data. Our models generalize the inverse Gaussian and one-sided stable autoregressive models existing in the literature. [ABSTRACT FROM AUTHOR]

We consider a stochastic nonlinear defocusing Schrödinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear noise in the Itô form. We work at the same time on compact Riemannian manifolds without boundary and on relatively compact smooth domains with either the Dirichlet or the Neumann boundary conditions, always in dimension two. We construct a martingale solution using a modified Faedo–Galerkin's method, following Brzeźniak et al (2019 Probab. Theory Relat. Fields 174 1273–338). Then by means of the Strichartz estimates deduced from Blair et al (2008 Proc. Am. Math. Soc. 136 247–56) but modified for our stochastic setting we show the pathwise uniqueness of solutions. Finally, we prove the existence of an invariant measure by means of a version of the Krylov–Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler (1999 Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 69–78). This is the first result of this type for stochastic nonlinear Schrödinger equation (NLS) on compact Riemannian manifolds without boundary and on relatively compact smooth domains even for an additive noise. Some remarks on the uniqueness in a particular case are provided as well. [ABSTRACT FROM AUTHOR]

Motivated by examples from extreme value theory, but without using the theory of regularly varying time series or any assumptions about the marginal distribution, we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. We explore general invariance properties of cluster processes that are implied by stationarity of the underlying time series. Of particular interest in applications are the cluster size distributions, and we derive general properties and interconnections between the size of an inspected and a typical cluster. While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a “mean cluster size”, we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory.

We apply the classical finite Fourier transform to construct an embedded functional process to a given multivariate time series model. The basic properties of the embedded functional process are presented. It appears that the embedded functional process is quite useful for the model building and prediction. We also provide a new method for approximating an infinite functional process with a finite functional process. The methodology is different from the classical one. Indeed, the active underlying frequencies are taken into consideration rather than the significant eigenvectors. The performance of our method is illustrated through simulations. Interestingly, in doing prediction, it appears that for our proposal the computation time is much shorter compared to existing ones. [ABSTRACT FROM AUTHOR]

This study introduces a novel stochastic variant for the Susceptible-Infected-Recovered-Susceptible (SIRS) system, focusing on perturbations involving the immunity decay rate. We determine a critical threshold value of the reproduction number, denoted as R 0 , which plays a pivotal role in understanding the system dynamics. Through rigorous mathematical derivations, we have shown that a unique solution exists for the system under consideration. Additionally, we leverage the powerful analytical tool of a stochastic Lyapunov function to evaluate the extinction and persistence of the infection, providing valuable insights into the system behavior under different conditions. Our analysis reveals that if R 0 < 1 , the disease will eventually vanish from the population, whereas if R 0 > 1 , an outbreak will ensue. To reinforce our findings, we provide computer simulations as supplementary evidence. [ABSTRACT FROM AUTHOR]

An apportionment paradox occurs when the rules for apportionment in a political system or distribution system produce results which seem to violate common sense. For example, The Alabama paradox occurs when the total number of seats increases but decreases the allocated number of a state and the population paradox occurs when the population of a state increases but its allocated number of seats decreases. The Balinski-Young impossibility theorem showed that there is no deterministic apportionment method that can avoid the violation of the quota rule and doesn’t have both the Alabama and the population paradoxes. In this paper, we propose a randomized apportionment method as a stochastic solution to the Balinski-Young impossibility. [ABSTRACT FROM AUTHOR]

In this paper, we study the discretization of the ergodic Functional Central Limit Theorem (CLT) established by Bhattacharya (see Bhattacharya in Z Wahrscheinlichkeitstheorie Verwandte Geb 60:185–201, 1982) which states the following: Given a stationary and ergodic Markov process (X t) t ⩾ 0 with unique invariant measure ν and infinitesimal generator A, then, for every smooth enough function f, (n 1 / 2 1 n ∫ 0 nt A f (X s) d s) t ⩾ 0 converges in distribution towards the distribution of the process (- 2 ⟨ f , A f ⟩ ν W t) t ⩾ 0 with (W t) t ⩾ 0 a Wiener process. In particular, we consider the marginal distribution at fixed t = 1 , and we show that when ∫ 0 n A f (X s) d s is replaced by a well chosen discretization of the time integral with order q (e.g. Riemann discretization in the case q = 1 ), then the CLT still holds but with rate n q / (2 q + 1) instead of n 1 / 2 . Moreover, our results remain valid when (X t) t ⩾ 0 is replaced by a q-weak order approximation (not necessarily stationary). This paper presents both the discretization method of order q for the time integral and the q-order ergodic CLT we derive from them. We finally propose applications concerning the first order CLT for the approximation of Markov Brownian diffusion stationary regimes with Euler scheme (where we recover existing results from the literature) and the second order CLT for the approximation of Brownian diffusion stationary regimes using Talay's scheme (Talay in Stoch Stoch Rep 29:13–36, 1990) of weak order two. [ABSTRACT FROM AUTHOR]

In the article Quenched large deviation principle for words in a letter sequence, Probab. Theory Relat. Fields 148, no. 3/4 (2010), 403–456 we derived a quenched large deviation principle for the empirical process of words obtained by cutting an i.i.d. sequence of letters according to an independent renewal process. We derived a representation of the associated rate function for stationary word processes in terms of certain specific relative entropies. Our proof of this representation is correct when the mean word length is finite, but is flawed when the mean word length is infinite. In this paper we fix the flaw in the proof. Along the way we derive new representations of the rate function that are interesting in their own right. A key ingredient in the proof is the observation that if the rate function in the annealed large deviation principle is finite at a stationary word process, then the letters in the tail of the long words in this process are typical. [ABSTRACT FROM AUTHOR]

This paper deals with inference in a class of stable but nearly-unstable processes. Autoregressive processes are considered, in which the bridge between stability and instability is expressed by a time-varying companion matrix A n with spectral radius ρ (A n) < 1 satisfying ρ (A n) → 1 . This framework is particularly suitable to understand unit root issues by focusing on the inner boundary of the unit circle. Consistency is established for the empirical covariance and the OLS estimation together with asymptotic normality under appropriate hypotheses when A, the limit of A n , has a real spectrum, and a particular case is deduced when A also contains complex eigenvalues. The asymptotic process is integrated with either one unit root (located at 1 or - 1 ), or even two unit roots located at 1 and - 1 . Finally, a set of simulations illustrate the asymptotic behavior of the OLS. The results are essentially proved by L 2 computations and the limit theory of triangular arrays of martingales. [ABSTRACT FROM AUTHOR]

The classical autoregressive type models are widely used in time series modelling. Recently, a class of models known as generalized autoregressive, recognized by an additional parameter, has been proposed in order to reveal some hidden features which cannot be characterized by the standard autoregressive models. In this paper, the generalized autoregressive models are extended to the vector-valued autoregressive models which provide a flexible framework for modelling the dependent data. The properties of the new model such as stationary conditions, some explicit form of the auto-covariance function and the spectral density matrices are investigated. Unknown parameters are then estimated and compared with other kinds of traditional methods. The numerical results obtained by means of simulation studies are then reported. Finally, the traditional autoregressive model and generalized autoregressive model are fitted to a well-known bivariate time series, respectively, and the performance of the proposed models and the estimation methods are discussed. [ABSTRACT FROM AUTHOR]

This paper presents new consistent methods for continuous-time Multiple-Input-Single-Output (MISO) Errors-In-Variables (EIV) systems by fractional models. The proposed idea is to use Higher-Order Statistics (HOS), such as fourth-order cumulants (foc), instead of noisy input and output measurements to obtain unbiased estimates. Firstly, all differentiation orders are assumed to be known a priori and linear coefficients are estimated. The developed estimator is based on minimizing the equation error and it is called fractional fourth-order based-least squares estimator ( f r a c - f o c - l s ). Secondly, the global commensurability of the fractional MISO system is considered. The f r a c - f o c - l s is combined with a non linear technique to estimate the global commensurate order along with linear coefficients. The developed algorithm is based on minimizing the output error and called fractional fourth-order cumulants based-least squares combined with global commensurate order optimization ( f r a c - f o c - g c o o l s ). The consistency of the developed estimators, in presence of high levels of noise corrupting both the input and output measurements, is assessed through a numerical example with the help of Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

Given a convex function Φ : [ 0 , 1 ] → R and the mean E f (X) = a ∈ [ 0 , 1 ] , which Boolean function f maximizes the Φ -stability E [ Φ (T ρ f (X)) ] of f? Here X is a random vector uniformly distributed on the discrete cube { - 1 , 1 } n and T ρ is the Bonami–Beckner operator. Special cases of this problem include the (symmetric and asymmetric) α -stability problems and the "Most Informative Boolean Function" problem. In this paper, we provide several upper bounds for the maximal Φ -stability. When specializing Φ to some particular forms, by these upper bounds, we partially resolve Mossel and O'Donnell's conjecture on α -stability with α > 2 , Li and Médard's conjecture on α -stability with 1 < α < 2 , and Courtade and Kumar's conjecture on the "Most Informative Boolean Function" which corresponds to a conjecture on α -stability with α = 1 . Our proofs are based on discrete Fourier analysis, optimization theory, and improvements of the Friedgut–Kalai–Naor (FKN) theorem. Our improvements of the FKN theorem are sharp or asymptotically sharp for certain cases. [ABSTRACT FROM AUTHOR]

U-statistics represent a fundamental class of statistics from modelling quantities of interest defined by multi-subject responses. U-statistics generalize the empirical mean of a random variable X to sums over every m-tuple of distinct observations of Stute (Ann. Probab. 19, 812–825 1991) introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: r (t) : = E [ φ (Y 1 , ... , Y m) | (X 1 , ... , X m) = t ] , for t ∈ ℝ d m. We apply the methods developed in Dony and Mason (Bernoulli 14(4), 1108–1133 2008) to establish uniform in t and in bandwidth consistency (i.e., h, h ∈ [an,bn] where 0 < a n < b n → 0 at some specific rate) to r(t) of the estimator proposed by Stute when Y, under weaker conditions on the kernel than previously used in the literature. We extend existing uniform bounds on the kernel conditional U-statistic estimator and make it adaptive to the intrinsic dimension of the underlying distribution of X which the so-called intrinsic dimension will characterize. In addition, uniform consistency is also established over φ ∈ 풡 for a suitably restricted class 풡 , in both cases bounded and unbounded, satisfying some moment conditions. Our theorems allow data-driven local bandwidths for these statistics. Moreover, in the same context, we show the uniform bandwidth consistency for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship, which is of its own interest. The theoretical uniform consistency results established in this paper are (or will be) key tools for many further developments in regression analysis. [ABSTRACT FROM AUTHOR]

Social media needs social networks to disseminate information among the people who like to interact with each other. Identification of the most prominent influencers is a crucial problem for influence diffusion in applications like viral marketing. Most of the previous works on influence diffusion studied the topological nature of the users but ignored the effects of communities within a network. This paper proposes a new method called the TSGC method, where the whole graph is divided into different non-overlapping communities. Each of the communities is taken as a subgraph by ignoring the connecting links between them. The most influential users are identified by using each node's local and global centrality measures in the given subgraph of a graph. Finally, the ranking of each node is performed by calculating the I s c o r e p of each node within a whole network. Experimental results on six datasets confirm that the proposed TSGC method outperforms many well-known existing methods in terms of influence diffusion phenomenon under both the LT and IC models. This paper also proposes a new model where an active promoter may lose his influencing potential over time, go to a recovered state where he is no longer active or can activate others, and then go to a susceptible state where he is prone to getting influenced in the future. A user who is influenced by an active user can also become an active user. We termed this model the ICIRS model. This model undergoes influence diffusion in continuous time, unlike discrete-time steps as focused in most of the existing papers. Our experimental evaluations on the datasets reveal that the ICIRS model performs non-progressive influence diffusion. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analytically at high dimensions, we also consider simple alternative entropy functionals, whose behavior, on the one hand, mimics the behavior of entropy and, on the other hand, is not difficult to study. Asymptotic behavior of the normalized entropy (so called entropy rate) is also studied for the entropy and for the alternative functionals. [ABSTRACT FROM AUTHOR]

The J-Bessel univariate kernel Ω d introduced by Schoenberg plays a central role in the characterization of stationary isotropic covariance models defined in a d-dimensional Euclidean space. In the multivariate setting, a matrix-valued isotropic covariance is a scale mixture of the kernel Ω d against a matrix-valued measure that is nondecreasing with respect to matrix inequality. We prove that constructions based on a p-variate kernel [ Ω d ij ] i , j = 1 p are feasible for different dimensions d ij , at the expense of some parametric restrictions. We illustrate how multivariate covariance models inherit such restrictions and provide new classes of hypergeometric, Matérn, Cauchy and compactly-supported models to illustrate our findings. [ABSTRACT FROM AUTHOR]

We study the limit behaviour of sequences of non-convex, vectorial, random integral functionals, defined on W 1 , 1 , whose integrands are ergodic and satisfy degenerate linear growth conditions. The latter involve suitable random, scale-dependent weight-functions. Under minimal assumptions on the integrand and on the weight-functions, we show that the sequence of functionals homogenizes to a non-degenerate deterministic functional defined on BV. [ABSTRACT FROM AUTHOR]

This paper deals with the rate of convergence for the central limit theorem of estimators of the drift coefficient, denoted θ, for the Ornstein-Uhlenbeck process X : = { X t , t ≥ 0 } observed at high frequency. We provide an approximate minimum contrast estimator and an approximate maximum likelihood estimator of θ, namely θ ˜ n : = 1 / (2 n ∑ i = 1 n X t i 2) , and θ ˆ n : = − ∑ i = 1 n X t i − 1 (X t i − X t i − 1 ) / (Δ n ∑ i = 1 n X t i − 1 2) , respectively, where t i = i Δ n , i = 0 , 1 , ... , n , Δ n → 0 . We provide Wasserstein bounds in the central limit theorem for θ ˜ n and θ ˆ n . [ABSTRACT FROM AUTHOR]

From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Is every point contained in balloons infinitely often or not? We answer this for the Euclidean space, the hyperbolic plane and regular trees. The result for the Euclidean space relies on a novel 0–1 law for stationary processes. Towards establishing the results for the hyperbolic plane and regular trees, we prove an upper bound on the density of any well-separated set in a regular tree which is a factor of an i.i.d. process. [ABSTRACT FROM AUTHOR]