On the functional central limit theorem via martingale approximation

Embedding of random walk, Brownian martingales, moment identities and estimates, rate of continuity, approximations of random walk, functional central limit theorem, laws of the iterated logarithm, arcsine laws, approximation of renewal processes, empirical distribution functions, embedding and approximation of martingales, martingales with small jumps, time-scale comparison.

We start by stating the need for a Gaussian approximation for dependent structures in the form of the central limit theorem (CLT) or of the functional CLT. To justify the need to quantify the dependence, we introduce illustrative examples: linear processes, functions of stationary sequences, recursive sequences, dynamical systems, additive functionals of Markov chains, and self-interactions. The limiting behavior of the associated partial sums can be handled with tools developed throughout the book. We also present basic notions for stationary sequences of random variables: various definitions and constructions, and definitions of ergodicity, projective decomposition, and spectral density. Special attention is given to dynamical systems, as many of our results also apply in this context. The chapter also surveys the basic theory of the convergence of stochastic processes in distribution, and introduces the reader to tightness, finite-dimensional convergence, and the need for maximal inequalities. It ends with the concepts of the moderate deviations principle and its functional form.

In this paper we study the Markov-modulated M/M/ $$\infty $$ queue, with a focus on the correlation structure of the number of jobs in the system. The main results describe the system’s asymptotic behavior under a particular scaling of the model parameters in terms of a functional central limit theorem. More specifically, relying on the martingale central limit theorem, this result is established, covering the situation in which the arrival rates are sped up by a factor N and the transition rates of the background process by $$N^\alpha $$ , for some $$\alpha >0$$ . The results reveal an interesting dichotomy, with crucially different behavior for $$\alpha >1$$ and $$\alpha <1$$ , respectively. The limiting Gaussian process, which is of the Ornstein–Uhlenbeck type, is explicitly identified, and it is shown to be in accordance with explicit results on the mean, variances and covariances of the number of jobs in the system.

In this paper we study the functional central limit theorem (CLT) for stationary Markov chains with a self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n, and establish that conditional convergence in distribution of partial sums implies the functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.

Convergence of averages of scaled functions of I(1) linear processes

A functional central limit theorem is proved for the number of solutions $(p,q)$ of the inequality $|q\omega - p| < f(q){q^{ - 1}},q \leq n$ (respectively $0 < q\omega - p < f(q){q^{ - 1}},q \leq n$ for some functions $f$ having a positive limit.

We study independent and identically distributed random iterations of continuous maps defined on a connected closed subset S of the Euclidean space Rk . We assume the maps are monotone (with respect to a suitable partial order) and a ‘topological’ condition on the maps. Then, we prove the existence of a pullback random attractor whose distribution is the unique stationary measure of the random iteration, and we obtain the synchronisation of random orbits. As a consequence of the synchronisation phenomenon, a functional central limit theorem is established.

In 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.

We study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.

The aim of this study is to extend the scope and applicability of the level-crossing method to discrete-time stochastic processes and generalize it to enable us to study multiple discrete-time stochastic processes. In previous versions of the level-crossing method, problems with it correspond to the fact that this method had been developed for analyzing a continuous-time process or at most a multiple continuous-time process in an individual manner. However, since all empirical processes are discrete in time, the already-established level-crossing method may not prove adequate for studying empirical processes. Beyond this, due to the fact that most empirical processes are coupled; their individual study could lead to vague results. To achieve the objectives of this study, we first find an analytical expression for the average frequency of crossing a level in a discrete-time process, giving the measure of the time experienced for two consecutive crossings named as the "waiting time." We then introduce the generalized level-crossing method by which the consideration of coupling between the components of a multiple process becomes possible. Finally, we provide an analytic solution when the components of a multiple stochastic process are independent Gaussian white noises. The comparison of the results obtained for coupled and uncoupled processes measures the strength and efficiency of the coupling, justifying our model and analysis. The advantage of the proposed method is its sensitivity to the slightest coupling and shortest correlation length.

In a state-dependent queueing network, arrival and service rates, as well as routing probabilities, depend on the vector of queue lengths. For properly normalized such networks, we derive functional laws of large numbers (FLLNs) and functional central limit theorems (FCLTs). The former support fluid approximations and the latter support diffusion refinements. The fluid limit in FLLN is the unique solution to a multidimensional autonomous ordinary differential equation with state-dependent reflection. The diffusion limit in FCLT is the unique strong solution to a stochastic differential equation with time-dependent reflection. Examples are provided that demonstrate how such approximations facilitate the design, analysis and optimization of various manufacturing, service, communication and other systems.

We take a Student process that is based on independent copies of a random variable \({X}\) and has trajectories in the function space \({D}\)[0, 1]. As a consequence of a functional central limit theorem (FCLT) for this process, with \({X}\) in the domain of attraction of the normal law, we consider convergence in distribution of five functionals of this process and derive respective asymptotic confidence intervals for the mean of \({X}\). We conclude that the obtained intervals have higher finite-sample coverage probabilities, or shorter expected lengths, than those of a classical asymptotic confidence interval, \({I_0}\), that follows simply from the asymptotic normality of the Student \({t}\)-statistic. Thus, the five FCLT based intervals may present reasonable alternatives to \({I_0}\).

Dealing with uncertainty about the true data generating process requires a differentiated perspective of the distributions in hypothesis testing. In particular, the realizations, or the observed data, generated by interactions that are naturally ordered in time, posits a need for a differentiated analysis with respect to the standard statistics available for hypothesis testing. The Functional Central Limit Theorem provides a framework that enables the researcher to build a statistic that fits his data and hypothesis at hand. In this paper I show some of the necessary conditions under which the popular t-statistic properly condenses the information of the underlying distribution as well as the additional tools available when the t distribution is not suitable for hypothesis testing.

Poisson processes (possibly nonhomogeneous) are constructed in the function spaces D q ≡D([0, 1] q , R) and Dq q x ⋯ x D q in order to approximate superpositions of uniformly sparse point processes and partial sums of infinitesimal integer-valued nonnegative random variables. Bounds for the Prohorov distance are computed, where the Prohorov distance is defined on the space of all probability measures on D q, with the Skorohod metric being used on D q. These bounds yield functional central limit theorems (invariance principles) and rates of convergence for functional central limit theorems involving convergence to the Poisson process. In this regard, this paper is an extension of Section 6 of Dudley [4].

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.