Functional Limit Theorem Research Articles

Limit theorems for additive functionals of random walks in random scenery

We study the asymptotic behaviour of additive functionals of random walks in random scenery. We establish bounds for the moments of the local time of the Kesten and Spitzer process. These bounds combined with a previous moment convergence result (and an ergodicity result) imply the convergence in distribution of additive observables (with a normalization in n1 4). When the sum of the observable is null, the previous limit vanishes and we prove the convergence in the sense of moments (with a normalization in n1 8).

Limit theorems for Lévy flights on a 1D Lévy random medium

We study a random walk on a point process given by an ordered array of points (ωk,k∈Z) on the real line. The distances ωk+1−ωk are i.i.d. random variables in the domain of attraction of a β-stable law, with β∈(0,1)∪(1,2). The random walk has i.i.d. jumps such that the transition probabilities between ωk and ωℓ depend on ℓ−k and are given by the distribution of a Z-valued random variable in the domain of attraction of an α-stable law, with α∈(0,1)∪(1,2). Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters α and β, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.

Limit theorems for trawl processes

In this work we derive limit theorems for trawl processes. First,we study the asymptotic behaviour of the partial sums of the discretized trawl process $(X_{i\Delta_{n}})_{i=0}^{\lfloor nt\rfloor-1}$, under the assumption that as $n\uparrow\infty$, $\Delta_{n}\downarrow0$ and $n\Delta_{n}\rightarrow\mu\in[0,+\infty]$. Second, we derive a functional limit theorem for trawl processes as the L\'{e}vy measure of the trawl seed grows to infinity and show that the limiting process has a Gaussian moving average representation.

Functional limit theorems for marked Hawkes point measures

This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian white noise process plus an appropriate lifting map of a correlated one-dimensional Brownian motion. The Brownian motion results from the self-exciting arrivals of events. We apply our limit theorems for Hawkes point measures to analyze the population dynamics of budding microbes in a host as well as their interaction with that host.

Asymptotic behaviour and functional limit theorems for a time changed Wiener process

We study the asymptotic behaviour of a properly normalized time changed Wiener processes. The time change reflects the fact that we consider the Laplace operator (which generates a Wiener process) multiplied by a possibly degenerate state-space dependent intensity λ(x). Applying a functional limit theorem for the superposition of stochastic processes, we prove functional limit theorems for the normalized time changed Wiener process. The normalization depends on the asymptotic behaviour of the intensity function λ. One of the possible limits is a skew Brownian motion.

Staffing many‐server queues with autoregressive inputs

AbstractRecent studies reveal significant overdispersion and autocorrelation in arrival data at service systems such as call centers and hospital emergency departments. These findings stimulate the needs for more practical non‐Poisson customer arrival models, and more importantly, new staffing formulas to account for the autocorrelative features in the arrival model. For this purpose, we study a multiserver queueing system where customer arrivals follow a doubly stochastic Poisson point process whose intensities are driven by a Cox–Ingersoll–Ross (CIR) process. The nonnegativity and autoregressive feature of the CIR process makes it a good candidate for modeling temporary dips and surges in arrivals. First, we devise an effective statistical procedure to calibrate our new arrival model to data which can be seen as a specification of the celebrated expectation–maximization algorithm. Second, we establish functional limit theorems for the CIR process, which in turn facilitate the derivation of functional limit theorems for our queueing model under suitable heavy‐traffic regimes. Third, using the corresponding heavy traffic limits, we asymptotically solve an optimal staffing problem subject to delay‐based constraints on the service levels. We find that, in order to achieve the designated service level, such an autoregressive feature in the arrival model translates into notable adjustment in the staffing formula, and such an adjustment can be fully characterized by the parameters of our new arrival model. In this respect, the staffing formulas acknowledge the presence of autoregressive structure in arrivals. Finally, we extend our analysis to queues having customer abandonment and conduct simulation experiments to provide engineering confirmations of our new staffing rules.

On stationarity properties of generalized Hermite-type processes

The paper investigates properties of generalized Hermite-type processes that arise in non-central limit theorems for integral functionals of long-range dependent random fields. The case of increasing multidimensional domain asymptotics is studied. Three approaches to investigate the properties of these processes are discussed. Contrary to the classical one-dimensional case, it is shown that for any choice of a multidimensional observation window the generalized Hermite-type process has non-stationary increments.

Functional Limit Theorems for the Fractional Ornstein\u2013Uhlenbeck Process

We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein–Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by both long- and short-range-dependent noise. The limit process has both Gaussian and non-Gaussian components. The theorem holds for any L^2 functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the Hölder topology, the rough topology for processes in C^{frac{1}{2}+}. This leads to a ‘rough creation’ / ‘rough homogenization’ theorem, by which we mean the weak convergence of a family of random smooth curves to a non-Markovian random process with non-differentiable sample paths. In particular, we obtain effective dynamics for the second-order problem and for the kinetic fractional Brownian motion model.

Multiple block sizes and overlapping blocks for multivariate time series extremes

Block maxima methods constitute a fundamental part of the statistical toolbox in extreme value analysis. However, most of the corresponding theory is derived under the simplifying assumption that block maxima are independent observations from a genuine extreme value distribution. In practice, however, block sizes are finite and observations from different blocks are dependent. Theory respecting the latter complications is not well developed, and, in the multivariate case, has only recently been established for disjoint blocks of a single block size. We show that using overlapping blocks instead of disjoint blocks leads to a uniform improvement in the asymptotic variance of the multivariate empirical distribution function of rescaled block maxima and any smooth functionals thereof (such as the empirical copula), without any sacrifice in the asymptotic bias. We further derive functional central limit theorems for multivariate empirical distribution functions and empirical copulas that are uniform in the block size parameter, which seems to be the first result of this kind for estimators based on block maxima in general. The theory allows for various aggregation schemes over multiple block sizes, leading to substantial improvements over the single block length case and opens the door to further methodology developments. In particular, we consider bias correction procedures that can improve the convergence rates of extreme-value estimators and shed some new light on estimation of the second-order parameter when the main purpose is bias correction.

A fundamental problem of hypothesis testing with finite inventory in e‐commerce

AbstractWe draw attention to a problem that is often overlooked or ignored by companies practicing hypothesis testing (A/B testing) in online environments. We show that conducting experiments on limited inventory that is shared between variants in the experiment can lead to high false‐positive rates since the core assumption of independence between the groups is violated. We provide a detailed analysis of the problem in a simplified setting whose parameters are informed by realistic scenarios. The setting we consider is a two‐dimensional (2D) random walk in a semiinfinite strip. It is rich enough to take a finite inventory into account, but is at the same time simple enough to allow for a closed form of the false‐positive probability. We prove that high false‐positive rates can occur, and develop tools that are suitable to help design adequate tests in follow‐up work. Our results also show that high false‐negative rates may occur. The proofs rely on a functional limit theorem for the 2D random walk in a semiinfinite strip.

Multidimensional Walks with Random Tendency

We introduce a multidimensional walk with memory and random tendency. The asymptotic behaviour is characterized, proving a law of large numbers and showing a phase transition from diffusive to superdiffusive regimes. In first case, we obtain a functional limit theorem to Gaussian vectors. In superdiffusive, we obtain strong convergence to a non-Gaussian random vector and characterize its moments.

Functional central limit theorems and moderate deviations for Poisson cluster processes

AbstractIn this paper, we consider functional limit theorems for Poisson cluster processes. We first present a maximal inequality for Poisson cluster processes. Then we establish a functional central limit theorem under the second moment and a functional moderate deviation principle under the Cramér condition for Poisson cluster processes. We apply these results to obtain a functional moderate deviation principle for linear Hawkes processes.

Ordinal patterns in long‐range dependent time series

AbstractWe analyze the ordinal structure of long‐range dependent time series. To this end, we use so called ordinal patterns which describe the relative position of consecutive data points. We provide two estimators for the probabilities of ordinal patterns and prove limit theorems in different settings, namely stationarity and (less restrictive) stationary increments. In the second setting, we encounter a Rosenblatt distribution in the limit. We prove more general limit theorems for functions with Hermite rank 1 and 2. We derive the limit distribution for an estimation of the Hurst parameter H if it is higher than 3/4. Thus, our theorems complement results for lower values of H which can be found in the literature. Finally, we provide some simulations that illustrate our theoretical results.

Functional limit theorem for the local time of stopped random walk

AbstractInteger random walk {Sn, n ≥ 0} with zero drift and finite variance σ2 stopped at the moment T of the first visit to the half axis (-∞, 0] is considered. For the random process which associates the variable u ≥ 0 with the number of visits the state ⌊uσ$\begin{array}{} \displaystyle \sqrt{n} \end{array}$⌋ by this walk conditioned on T > n, the functional limit theorem on the convergence to the local time of stopped Brownian meander is proved.

A functional limit theorem for nearly critical branching process with immigration

A functional limit theorem for nearly critical branching process with immigration

On the Estimation in Multi-server Multi-Core Open Queuing Networks

The paper is devoted to the analysis of queueing systems in the context of the network and communications theory. We investigate the estimation in a multi-server multi-core open queueing networks and its applications to the theorems in heavy traffic conditions (fluid approximation, functional limit theorem, and law of the iterated logarithm) for a queue of jobs in a multi-server multi-core open queueing networks..

Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.

Limit theorems for additive functionals of continuous time random walks

AbstractFor a continuous-time random walk X = {Xt, t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$, t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.

Functional Central Limit Theorems and P(ϕ)1-Processes for the Relativistic and Non-Relativistic Nelson Models

We construct P(ϕ)1-processes indexed by the full time-line, separately derived from the functional integral representations of the relativistic and non-relativistic Nelson models in quantum field theory. These two cases differ essentially by sample path regularity. Associated with these processes we define a martingale which, under an appropriate scaling, allows to obtain a central limit theorem for additive functionals of these processes. We discuss a number of examples by choosing specific functionals related to particle-field operators.

On nonparametric ridge estimation for multivariate long-memory processes

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.