A Functional Central Limit Theorem in Diophantine Approximation

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

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.

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.

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 \({I_0}\).

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.

Limit theorems for cumulative processes

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].

In this paper, we develop necessary and sufficient conditions for the\nvalidity of a martingale approximation for the partial sums of a stationary\nprocess in terms of the maximum of consecutive errors. Such an approximation is\nuseful for transferring the conditional functional central limit theorem from\nthe martingale to the original process. The condition found is simple and well\nadapted to a variety of examples, leading to a better understanding of the\nstructure of several stochastic processes and their asymptotic behaviors. The\napproximation brings together many disparate examples in probability theory. It\nis valid for classes of variables defined by familiar projection conditions\nsuch as the Maxwell--Woodroofe condition, various classes of mixing processes,\nincluding the large class of strongly mixing processes, and for additive\nfunctionals of Markov chains with normal or symmetric Markov operators.\n

: Suppose that X is a positive recurrent Harris chain with invariant measure pi. We develop a Lyapunov function criterion that permits one to bound the solution g to Poisson's equation for X. This bound is then applied to obtain sufficient conditions that guarantee that the solution be an element of L sub p (pi). When p = 2, the square integrability of g implies the validity of a functional central limit theorem for the Markov chain. We illustrate the technique with applications to the waiting time sequence of the single-server queue and autoregressive sequences. Keywords: Functional central limit theorem.

Under suitable physically reasonable initial assumptions, a functional central limit theorem is obtained for a nonequilibrium model of randomly interacting particles with unbounded jump intensity. This model is related to a nonlinear Boltzmann-type equation.