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In this paper, we discuss the difference between the deterministic and stochastic models of nonlocal diffusion. We use a nonlocal reaction-diffusion equation and a multi-dimensional jump Markov process to analyze these mathematical models. First, we demonstrate that the difference converges to 0 in probability with a supremum norm for a sizeable network. Next, we consider the rescaled difference and show that it converges to a stochastic process in distribution on the Skorokhod space. [ABSTRACT FROM AUTHOR]

We provide a functional central limit theorem for a broad class of smooth functions for possibly non-causal multivariate linear processes with time-varying coefficients. Since the limiting processes depend on unknown quantities, we propose a local block bootstrap procedure to circumvent this inconvenience in practical applications. In particular, we prove bootstrap validity for a very large class of processes. Our results are illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

We consider the partial-sum process ∑ k = 1 n t X k n , where X k n = ∑ j = 0 ∞ α j n ξ k - j b n , k ∈ Z , n ≥ 1, is a series of linear processes with tapered filter α j n = α j 1 0 ≤ j ≤ λ n and heavy-tailed tapered innovations ξj(b(n)), j ∈ Z. Both tapering parameters b(n) and ⋋ (n) grow to ∞ as n→∞. The limit behavior of the partial-sum process (in the sense of convergence of finite-dimensional distributions) depends on the growth of these two tapering parameters and dependence properties of a linear process with nontapered filter ai, i ≥ 0, and nontapered innovations. We consider the cases where b(n) grows relatively slowly (soft tapering) and rapidly (hard tapering) and all three cases of growth of ⋋(n) (strong, weak, and moderate tapering). [ABSTRACT FROM AUTHOR]

This paper develops a Gaussian model for an open network of queues having a path-dependent net-input process, whose evolution depends on its early history, and satisfies a non-ergodic law of large numbers. We show that the Gaussian model arises as the heavy-traffic limit for a sequence of open queueing networks, each with a multivariate generalization of a Polya arrival process. We show that the net-input and queue-length processes for the Gaussian model satisfy non-ergodic laws of large numbers with tractable distributions. [ABSTRACT FROM AUTHOR]

A scheduled arrival model is one in which the jth customer is scheduled to arrive at time jh but the customer actually arrives at time j h + ξ j , where the ξ j 's are independent and identically distributed. It has previously been shown that the arrival counting process for scheduled traffic obeys a functional central limit theorem (FCLT) with fractional Brownian motion (fBM) with Hurst parameter H ∈ (0 , 1 / 2) when the ξ j 's have a Pareto-like tail with tail exponent lying in (0, 1). Such limit processes exhibit less variability than Brownian motion, because the scheduling feature induces negative correlations in the arrival process. In this paper, we show that when the tail of the ξ j 's has a super-heavy tail, the FCLT limit process is Brownian motion (i.e., H = 1 / 2 ), so that the heaviness of the tails eliminates any remaining negative correlations and generates a limit process with independent increments. We further study the case when the ξ j 's have a Cauchy-like tail, and show that the limit process in this setting is a fBM with H = 0 . So, this paper shows that the entire range of fBMs with H ∈ [ 0 , 1 / 2 ] are possible as limits of scheduled traffic. [ABSTRACT FROM AUTHOR]

We prove the Rosnethal's inequality of m-dependent random variables under the sub-linear expectation in this paper. Furthermore, we use this inequality to investigate the central limit theorem for linear processes generated by m-dependent random variables under sub-linear expectations. This article use the basic definitions of sub-linear expectation space, Kronecker lemma, Cr inequality etc. to demonstrate the main conclusion. [ABSTRACT FROM AUTHOR]

We consider precise deviations for discrete ensembles. For β = 2 case, we first establish an asymptotic formula of the Christoffel–Darboux kernel of the discrete orthogonal polynomials on an infinite regular lattice with weight e−NV(x). Then we use the asymptotic formula to get the precise deviations of the extreme value for corresponding ensemble. [ABSTRACT FROM AUTHOR]

In this paper, we derive a self-normalized functional limit theorem for strictly stationary linear processes with i.i.d. heavy-tailed innovations and random coefficients under the condition that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series. The convergence takes place in the space of càdlàg functions on [0, 1] with the Skorokhod M2 topology. [ABSTRACT FROM AUTHOR]

We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size ρ may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive n log n scaling (i) for fixed infinite horizon configurations—letting first n → ∞ and then ρ → 0 —studied e.g. by Szász and Varjú (J Stat Phys 129(1):59–80, 2007) and (ii) Boltzmann–Grad type situations—letting first ρ → 0 and then n → ∞ —studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016). [ABSTRACT FROM AUTHOR]

A large deviation principle (LDP) for a class of additive functionals of semi-Markov processes and their associated Markov renewal processes is studied via an almost sure functional central limit theorem. The rate function corresponding to the deviations from the paths of the corresponding empirical processes with logarithmic averaging is determined as a relative entropy with respect to the Wiener measure on D [ 0 , ∞). A martingale decomposition for additive functionals of Markov renewal processes is employed. [ABSTRACT FROM AUTHOR]

The aim of this treatise is to introduce the concept of strongly almost convergence in complex uncertain sequences. Also, we investigate the strongly almost convergence in almost surely (in shortly a.s.), convergence in measure, convergence in mean, convergence in distribution and convergence in uniformly almost surely of complex uncertain sequences and study the relationships among them. [ABSTRACT FROM AUTHOR]

In this paper, we consider discrete time approximations for stochastic differential equations with the form: X t = X 0 + ∫ 0 t f (X s) d h s + ∫ 0 t g (X s) d Y s H , t > 0 , where h : R + → R is a continuous function with locally bounded variation, f , g : R → R are measurable functions, and the integral with respect to Y t H = ∫ 0 t σ s d S s H is the pathwise Riemann-Stieltjes integral, SH is a subfractional Brownian motion with H ∈ (1 2 , 1) , σ is a deterministic (possibly discontinuous) function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate functional limit problem for path of a Brownian sheet, Chung's functional law of the iterated logarithm for a Brownian sheet is obtained. The main tool in the proof is large deviation and small deviation for a Brownian sheet. [ABSTRACT FROM AUTHOR]

We prove a multivariate functional version of de Jong's CLT (J Multivar Anal 34(2):275–289, 1990) yielding that, given a sequence of vectors of Hoeffding-degenerate U-statistics, the corresponding empirical processes on [0, 1] weakly converge in the Skorohod space as soon as their fourth cumulants in t = 1 vanish asymptotically and a certain strengthening of the Lindeberg-type condition is verified. As an application, we lift to the functional level the 'universality of Wiener chaos' phenomenon first observed in Nourdin et al. (Ann Probab 38(5):1947–1985, 2010). [ABSTRACT FROM AUTHOR]

The median test has been proven to be more powerful than the Student t-test and the Wilcoxon–Mann–Whitney test in heavy-tailed cases for univariate data. The multivariate extension of the median test, for multidimensional data, was demonstrated to be more efficient than the Hotelling T 2 and the Wilcoxon–Mann–Whitney tests for high dimensions and in very heavy-tailed cases. On the basis of these postulates, in this paper, we construct a median-type test based on spatial ranks for functional data, i.e. in infinite-dimensional space, and we obtain asymptotic results. Then, we compare the proposed functional median test with numerous competing tests using simulated and real functional data: as in the univariate and multivariate cases, the proposed test is more adapted to heavy-tailed distributions. [ABSTRACT FROM AUTHOR]

The paper is devoted to the analysis of queueing systems in the context of the network and communication theory (called a multi-server multi-core open queueing network). The ob ject of this research on the queueing theory is theorems about the Functional Strong Laws of Large Numbers (FSLLN) in multi-server multi-core open queueing networks, working under overload heavy traffic conditions. FSLLN is known as a fluid limit or fluid approximation. In this paper, FSLLN are proved for extreme values of important probabilistic characteristics of the multi-server multicore open queueing network, investigated as well as the virtual waiting time of a job and the queue length of jobs. As applications of the proved theorems Little's laws in a multi-server multi-core open queueing network are presented. [ABSTRACT FROM AUTHOR]

The moving block bootstrap can be used to determine critical values for test statistics used to detect a change-point in the marginal distribution of a stationary time series. We examine the impact of incorporating an estimator of the change-point when centering the bootstrap blocks and establish conditions under which the bootstrapped test statistics remain stochastically bounded regardless of whether or not a change is present. [ABSTRACT FROM AUTHOR]

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on Z d . In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment. [ABSTRACT FROM AUTHOR]

We give an invariance principle for very general additive functionals of conditioned Bienaymé-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable Lévy tree. This includes the case when the offspring distribution has finite variance (the Lévy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height. [ABSTRACT FROM AUTHOR]

This paper is devoted to the asymptotic analysis of the reinforced elephant random walk (RERW) using a martingale approach. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the RERW. The distributional convergences of the RERW to some Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the RERW. All our analysis relies on asymptotic results for multi-dimensional martingales with matrix normalization. [ABSTRACT FROM AUTHOR]

The object of this research on queueing theory is to analyze the behaviour of open queueing network, working under overload heavy traffic conditions. We have proved probability limit theorem for the global values of virtual waiting time of a customer in open queueing networks. Finally, we present application of the recurrent method in the further analysis of open queuing networks. [ABSTRACT FROM AUTHOR]

In this paper, we study optimal control problems for multiclass G I / M / n + M queues in an alternating renewal (up–down) random environment in the Halfin–Whitt regime. Assuming that the downtimes are asymptotically negligible and only the service processes are affected, we show that the limits of the diffusion-scaled state processes under non-anticipative, preemptive, work-conserving scheduling policies, are controlled jump diffusions driven by a compound Poisson jump process. We establish the asymptotic optimality of the infinite-horizon discounted and long-run average (ergodic) problems for the queueing dynamics. Since the process counting the number of customers in each class is not Markov, the usual martingale arguments for convergence of mean empirical measures cannot be applied. We surmount this obstacle by demonstrating the convergence of the generators of an augmented Markovian model which incorporates the age processes of the renewal interarrival times and downtimes. We also establish long-run average moment bounds of the diffusion-scaled queueing processes under some (modified) priority scheduling policies. This is accomplished via Foster–Lyapunov equations for the augmented Markovian model. [ABSTRACT FROM AUTHOR]

In the paper, we consider the limit behavior of partial-sum random field (r.f.) S n t 1 t 2 X b n = ∑ k = 1 n 1 t 1 ∑ l = 1 n 2 t 2 X k , l b n , where X k , l b n = ∑ i = 0 ∞ ∑ j = 0 ∞ c i , j ξ k − i , l − j b n k l ∈ ℤ , n ≥ 1 , is a family (indexed by n = (n1, n2), ni ≥ 1) of linear r.f.s with filter ci,j = aibj and innovations ξk,l(b(n)) having heavy-tailed tapered distributions with tapering parameter b(n) growing to infinity as n → ∞. In [V. Paulauskas, Limit theorems for linear random fields with tapered innovations. I: The Gaussian case, Lith. Math. J., 61(2):261–273, 2021], we considered the so-called hard tapering as b(n) grows relatively slowly and the limit r.f.s for appropriately normalized Sn(t1, t2;X(b(n))) are Gaussian. In this paper, we consider the case of soft tapering where b(n) grows more rapidly in comparison with the case of hard tapering and stable limit r.f.s.We consider cases where the sequences {ai} and {bj} are long-range, short-range, and negatively dependent. [ABSTRACT FROM AUTHOR]

Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes—the random wind-tree model introduced in Ehrenfest–Ehrenfest (1912) as reported by Ehrenfest, Ehrenfest (Begriffliche Grundlagen der statistischen Auffassung in der Mechanik Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (Translated:) The conceptual foundations of the statistical approach in mechanics. Dover Books on Physics, 1912). We show that in the joint Boltzmann–Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to Brownian motion in a particular scaling limit. In a previous paper (2020) (Lutsko, Tóth in Commun. Math. Phys. 379:589–632, 2020) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, a similar strategy with some modification can be used to prove the invariance principle for the random wind-tree model. The key differences from our previous work are that the individual path segments of the underlying Markov process are no longer fully independent and the geometry of recollisions is simpler. [ABSTRACT FROM AUTHOR]

Let u = {u(t,x),t ∈ [0,T],x ∈ ℝ} be a solution to a stochastic heat equation driven by a space-time white noise. We study that the realized power variation of the process u with respect to the time, properly normalized, has Gaussian asymptotic distributions. In particular, we study the realized power variation of the process u with respect to the time converges weakly to Brownian motion. [ABSTRACT FROM AUTHOR]

We introduce multi-type Markov Branching trees, which are simple random population tree models where individuals are characterized by their size and their type and give rise to (size,type)-children in a Galton–Watson fashion, with the rule that the size of any individual is at least the sum of the sizes of its children. Assuming that the macroscopic size-splittings are rare, we describe the scaling limits of multi-type Markov Branching trees in terms of multi-type self-similar fragmentation trees. We observe three different regimes according to whether the probability of type change of a size-biased child is proportional to the probability of macroscopic splitting (the critical regime, in which we get in the limit multi-type fragmentation trees with indeed several types), smaller than the probability of macroscopic splitting (the solo regime, in which the limit trees are monotype as we never see a type change), or larger than the probability of macroscopic splitting (the mixing regime, in which case the types mix in the limit and we get monotype fragmentation trees). This framework allows us to unify models which may a priori seem quite different, a strength which we illustrate with two notable applications. The first one concerns the description of the scaling limits of growing models of random trees built by gluing at each step on the current structure a finite tree picked randomly in a finite alphabet of trees, extending Rémy's well-known algorithm for the generation of uniform binary trees to a fairly broad framework. We are then either in the critical regime with multi-type fragmentation trees in the scaling limit, or in the solo regime. The second application concerns the scaling limits of large multi-type critical Galton–Watson trees when the offspring distributions all have finite second moments. This topic has already been studied but our approach gives a different proof and we improve on previous results by relaxing some hypotheses. We are then in the mixing regime: the scaling limits are always multiple of the Brownian CRT, a pure monotype fragmentation tree in our framework. [ABSTRACT FROM AUTHOR]

We study random walks on Z d (with d ≥ 2 ) among stationary ergodic random conductances { C x , y : x , y ∈ Z d } that permit jumps of arbitrary length. Our focus is on the quenched invariance principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat-kernel technology assuming that the p-th moment of ∑ x ∈ Z d C 0 , x | x | 2 and q-th moment of 1 / C 0 , x for x neighboring the origin are finite for some p , q ≥ 1 with p - 1 + q - 1 < 2 / d . In particular, a QIP thus holds for random walks on long-range percolation graphs with connectivity exponents larger than 2d in all d ≥ 2 , provided all the nearest-neighbor edges are present. Although still limited by moment conditions, our method of proof is novel in that it avoids proving everywhere-sublinearity of the corrector. This is relevant because we show that, for long-range percolation with exponents between d + 2 and 2d, the corrector exists but fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in d ≥ 3 under the conditions complementary to those of the recent work of Bella and Schäffner (Ann Probab 48(1):296–316, 2020). These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems. [ABSTRACT FROM AUTHOR]

We consider the empirical two-sample U-statistic with β -mixing strictly stationary data and investigate its convergence in Skorohod spaces. We then provide an application of such convergence. [ABSTRACT FROM AUTHOR]

We prove a quenched invariance principle for a class of random walks in random environment on Z d , where the walker alters its own environment. The environment consists of an outgoing edge from each vertex. The walker updates the edge e at its current location to a new random edge e ′ (whose law depends on e) and then steps to the other endpoint of e ′ . We show that a native environment for these walks (i.e., an environment that is stationary in time from the perspective of the walker) consists of the wired uniform spanning forest oriented toward the walker, plus an independent outgoing edge from the walker. [ABSTRACT FROM AUTHOR]

In the framework of nonadditive probabilities or sublinear expectations, the Kolmogorov's strong law of large numbers (SLLN) states that for a sequence of independent and identically distributed (IID) random variables, limit points of its sample mean quasi-surely fall inside an interval given by a pair of lower and upper means. In this article, we will investigate a cluster set of limit points of a sequence of stochastic processes, which are given by linear interpolating of the sample mean of IID random variables under sublinear expectations, and show an invariance principle. The invariance principle will strengthen the Kolmogorov's SLLN under nonadditive probabilities in some extent. [ABSTRACT FROM AUTHOR]

In this note, we establish a functional central limit theorem for the capacity of the range for a class of α -stable random walks on the integer lattice Z d with d > 5 α / 2 . Using similar methods, we also prove an analogous result for the cardinality of the range when d > 3 α / 2 . [ABSTRACT FROM AUTHOR]

In this paper, we consider a multiplicative Poisson shot noise process with compensation which converges weakly to a fractional Brownian motion. Under mild conditions, the sample path large deviations for this process are established under the uniform topology. The methods include the weak convergence techniques, exponentially good approximation and exponential inequalities [ABSTRACT FROM AUTHOR]

In this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter ε such that, for every fixed value of the slow variable, the fast dynamics are sufficiently chaotic with ergodic invariant measure. Convergence of the slow process to the solution of a homogenized stochastic differential equation (SDE) in the limit ε to zero, with explicit formulas for drift and diffusion coefficients, has so far only been obtained for the case that the fast dynamics evolve independently. In this paper we give sufficient conditions for the convergence of the first moments of the slow variable in the coupled case. Our proof is based upon a new method of stochastic regularization and functional-analytical techniques combined via a double limit procedure involving a zero-noise limit as well as considering ε to zero. We also give exact formulas for the drift and diffusion coefficients for the limiting SDE. As a main application of our theory, we study weakly-coupled systems, where the coupling only occurs in lower time scales. [ABSTRACT FROM AUTHOR]

A functional law of the iterated logarithm (LIL) and its corresponding LIL are established for a multiclass single-server queue with first come first served (FCFS) service discipline. The functional LIL and its LIL quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. The functional LIL and LIL are established in three cases: underloaded, critically loaded and overloaded, for performance measures including the total workload, idle time, queue length, workload, busy time, departure and sojourn time processes. The proofs of the functional LIL and LIL are based on a strong approximation approach, which approximates discrete performance processes with reflected Brownian motions. Numerical examples are considered to provide insights on these limit results. [ABSTRACT FROM AUTHOR]

We establish a quenched local central limit theorem for the dynamic random conductance model on Z d only assuming ergodicity with respect to space-time shifts and a moment condition. As a key analytic ingredient we show Hölder continuity estimates for solutions to the heat equation for discrete finite difference operators in divergence form with time-dependent degenerate weights. The proof is based on De Giorgi's iteration technique. In addition, we also derive a quenched local central limit theorem for the static random conductance model on a class of random graphs with degenerate ergodic weights. [ABSTRACT FROM AUTHOR]

We consider the limit behavior of partial-sum random field S n t 1 t 2 X b n = ∑ k = 1 n 1 t 1 ∑ l = 1 n 2 t 2 X k , l n , where X k , l n = ∑ i = 0 ∞ ∑ j = 0 ∞ c i , j ξ k − i , l − j b n k l ∈ ℤ , n ≥ 1 , is a family (indexed by n = (n1, n2), ni ≥ 1) of linear random fields with filter ci,j = aibj and innovations ξk,l(bn) having heavy-tailed tapered distributions with tapering parameter bn growing to infinity as n→ ∞. We consider the so-called hard tapering as bn grows relatively slowly and the limit random fields for appropriately normalized Sn(t1, t2;X(bn)) are Gaussian random fields. We consider all cases where sequences {ai} and {bj} are long-range, short-range, and negative dependent. In the second part (as a separate paper), we will consider the case of soft tapering, where bn grows more rapidly, and limit random fields are stable. [ABSTRACT FROM AUTHOR]

In the paper, we consider some piecewise deterministic Markov process, whose continuous component evolves according to semiflows, which are switched at the jump times of a Poisson process. The associated Markov chain describes the states of this process directly after the jumps. Certain ergodic properties of these two dynamical systems have been already investigated in our recent papers. We now aim to establish the law of the iterated logarithm for the aforementioned continuous-time process. Moreover, we intend to do this using the already proven properties of the discrete-time system. The abstract model under consideration has interesting interpretations in real-life sciences, such as biology. Among others, it can be used to describe the stochastic dynamics of gene expression. [ABSTRACT FROM AUTHOR]

We study a continuous-time random walk on Z d in an environment of random conductances taking values in (0 , ∞) . For a static environment, we extend the quenched local limit theorem to the case of a general speed measure, given suitable ergodicity and moment conditions on the conductances and on the speed measure. Under stronger moment conditions, an annealed local limit theorem is also derived. Furthermore, an annealed local limit theorem is exhibited in the case of time-dependent conductances, under analogous moment and ergodicity assumptions. This dynamic local limit theorem is then applied to prove a scaling limit result for the space-time covariances in the Ginzburg–Landau ∇ ϕ model. We also show that the associated Gibbs distribution scales to a Gaussian free field. These results apply to convex potentials for which the second derivative may be unbounded. [ABSTRACT FROM AUTHOR]

A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability p ∈ (0 , 1) , U ^ n + 1 is sampled uniformly from U ^ 1 , ... , U ^ n , and with complementary probability 1 - p , U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when p < 1 / 2 , and that a further rescaling is needed when p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks. [ABSTRACT FROM AUTHOR]

This article investigates general scaling settings and limit distributions of functionals of filtered random fields. The filters are defined by the convolution of non-random kernels with functions of Gaussian random fields. The case of long-range dependent fields and increasing observation windows is studied. The obtained limit random processes are non-Gaussian. Most known results on this topic give asymptotic processes that always exhibit non-negative auto-correlation structures and have the self-similar parameter H ∈ (1 2 , 1) . In this work, we also obtain convergence for the case H ∈ (0 , 1 2) and show how the Hurst parameter H can depend on the shape of the observation windows. Various examples are presented. [ABSTRACT FROM AUTHOR]

In this article, we construct two families of processes, from a unique Lévy process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As applications of this result, we obtain families of processes that converge in law towards fractional Brownian motion, sub-fractional Brownian motion and bifractional Brownian motion, respectively. [ABSTRACT FROM AUTHOR]

For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index α ∈ (0 , 2) and weak dependence conditions. The limiting process consists of an α-stable Lévy process and an extremal process. We also describe the dependence between these two components of the limit. The convergence takes place in the space of R 2 -valued càdlàg functions on [ 0 , 1 ] , with the Skorohod weak M 1 topology. We further show that this topology in general can not be replaced by the stronger (standard) M 1 topology. [ABSTRACT FROM AUTHOR]

Donsker's theorem is perhaps the most famous invariance principle result for Markov processes. It states that, when properly normalized, a random walk behaves asymptotically like a Brownian motion. This approach can be extended to general Markov processes whose driving parameters are taken to a limit, which can lead to insightful results in contexts like large distributed systems or queueing networks. The purpose of this paper is to assess the rate of convergence in these so-called diffusion approximations, in a queueing context. To this end, we extend the functional Stein method, introduced for the Brownian approximation of Poisson processes, to two simple examples: the single-server queue and the infinite-server queue. By doing so, we complete the recent applications of Stein's method to queueing systems, with results concerning the whole trajectory of the considered process, rather than its stationary distribution. [ABSTRACT FROM AUTHOR]

Let ξ : Ω × R n → R be zero mean, mean-square continuous, stationary, Gaussian random field with covariance function r (x) = E [ ξ (0) ξ (x) ] and let G : R → R such that G is square integrable with respect to the standard Gaussian measure and is of Hermite rank d. The Breuer-Major theorem in it's continuous setting gives that, if r ∈ L d (R n) , then the finite dimensional distributions of Z s (t) = 1 (2 s) n / 2 ∫ [ − s t 1 / n , s t 1 / n ] n [ G (ξ (x)) − E [ G (ξ (x)) ] ] d x converge to that of a scaled Brownian motion as s → ∞. Here we give a proof for the case when ξ : Ω × R n → R m is a random vector field. We also give a proof for the functional convergence in C ([ 0 , ∞)) of Zs to hold under the condition that for some p > 2, G ∈ L p (R m , γ m) where γm denotes the standard Gaussian measure on R m and we derive expressions for the asymptotic variance of the second chaos component in the Wiener chaos expansion of Z s (1). [ABSTRACT FROM AUTHOR]

The K sample problem for high-dimensional vector time series is studied, especially focusing on sensor data streams, in order to analyze the second moment structure and detect changes across samples and/or across variables cumulated sum (CUSUM) statistics of bilinear forms of the sample covariance matrix. In this model, K independent vector time series Y T , 1 , ... , Y T , K are observed over a time span [ 0 , T ] , which may correspond to K sensors (locations) yielding d-dimensional data as well as K locations where d sensors emit univariate data. Unequal sample sizes are considered as arising when the sampling rate of the sensors differs. We provide large-sample approximations and two related change point statistics, a sum of squares and a pooled variance statistic. The resulting procedures are investigated by simulations and illustrated by analyzing a real data set. [ABSTRACT FROM AUTHOR]

We consider nonparametric estimation of the ridge of a probability density function for multivariate linear processes with long-range dependence. We derive functional limit theorems for estimated eigenvectors and eigenvalues of the Hessian matrix. We use these results to obtain the weak convergence for the estimated ridge and asymptotic simultaneous confidence regions. [ABSTRACT FROM AUTHOR]

This paper develops almost sure convergence for sums of negatively superadditive dependent random vectors in Hilbert spaces, we obtain Chung type SLLN and the Jaite type SLLN for sequences of negatively superadditive dependent random vectors in Hilbert spaces. Rate of convergence is studied through considering almost sure convergence to 0 of tail series. As an application, the almost sure convergence of degenerate von Mises-statistics is investigated. [ABSTRACT FROM AUTHOR]

The article focuses on the assumption of independence in various statistical models is only an approximation as it is often not valid for various real data. Topics include the covariance and spectral functions being commonly used to characterise the dependence properties of random fields and processes, the literature shows diverse definitions of long-range dependent random fields, and the random fields such as asymptotic of the covariance functions at infinity or spectral densities at zero.

How long does it take for an initially advantageous mutant to establish itself in a resident population, and what does the population composition look like then? We approach these questions in the framework of the so called Bare Bones evolution model (Klebaner et al. in J Biol Dyn 5(2):147–162, 2011. https://doi.org/10.1080/17513758.2010.506041) that provides a simplified approach to the adaptive population dynamics of binary splitting cells. As the mutant population grows, cell division becomes less probable, and it may in fact turn less likely than that of residents. Our analysis rests on the assumption of the process starting from resident populations, with sizes proportional to a large carrying capacity K. Actually, we assume carrying capacities to be a 1 K and a 2 K for the resident and the mutant populations, respectively, and study the dynamics for K → ∞ . We find conditions for the mutant to be successful in establishing itself alongside the resident. The time it takes turns out to be proportional to log K . We introduce the time of establishment through the asymptotic behaviour of the stochastic nonlinear dynamics describing the evolution, and show that it is indeed 1 ρ log K , where ρ is twice the probability of successful division of the mutant at its appearance. Looking at the composition of the population, at times 1 ρ log K + n , n ∈ Z + , we find that the densities (i.e. sizes relative to carrying capacities) of both populations follow closely the corresponding two dimensional nonlinear deterministic dynamics that starts at a random point. We characterise this random initial condition in terms of the scaling limit of the corresponding dynamics, and the limit of the properly scaled initial binary splitting process of the mutant. The deterministic approximation with random initial condition is in fact valid asymptotically at all times 1 ρ log K + n with n ∈ Z . [ABSTRACT FROM AUTHOR]

This paper is concerned with finite sample approximations to the supremum of a non-degenerate U-process of a general order indexed by a function class. We are primarily interested in situations where the function class as well as the underlying distribution change with the sample size, and the U-process itself is not weakly convergent as a process. Such situations arise in a variety of modern statistical problems. We first consider Gaussian approximations, namely, approximate the U-process supremum by the supremum of a Gaussian process, and derive coupling and Kolmogorov distance bounds. Such Gaussian approximations are, however, not often directly applicable in statistical problems since the covariance function of the approximating Gaussian process is unknown. This motivates us to study bootstrap-type approximations to the U-process supremum. We propose a novel jackknife multiplier bootstrap (JMB) tailored to the U-process, and derive coupling and Kolmogorov distance bounds for the proposed JMB method. All these results are non-asymptotic, and established under fairly general conditions on function classes and underlying distributions. Key technical tools in the proofs are new local maximal inequalities for U-processes, which may be useful in other problems. We also discuss applications of the general approximation results to testing for qualitative features of nonparametric functions based on generalized local U-processes. [ABSTRACT FROM AUTHOR]