Functional and random central limit theorems for the Robbins-Munro process

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

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.

Limit theorems for cumulative processes

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.

In this paper, we study the asymptotic behavior of a fully-coupled slow-fast McKean-Vlasov stochastic system. Using the non-linear Poisson equation on Wasserstein space, we first establish the strong convergence in the averaging principle of the functional law of large numbers type. In particular, the diffusion coefficient of the slow process can depend on the distribution of the fast motion. Then we consider the stochastic fluctuations of the original system around its average, and prove that the normalized difference will converge weakly to a linear McKean-Vlasov Ornstein-Uhlenbeck type process, which can be viewed as a functional central limit theorem. Extra drift and diffusion coefficients involving the expectation are characterized explicitly. Furthermore, the optimal rates of the convergence are also obtained.

Central limit theorems guarantee that the distributions of properly normalized sums of certain random variables are approximately normal. In many cases, however, a more detailed analysis is necessary. When testing for structural constancy in models, we might be interested in the temporal evolution of our sums. So for random variables X i we are interested in analysing the behaviour of as a function of t for t≤N. It is convenient to normalize the time, too, and consider for 0≤z≤1.

A central limit theorem and a functional central limit theorem are obtained for weighted linear process ofρ-mixing sequences for theXt=∑i=0∞aiYt−i, where{Yi,0≤i&lt;∞}is a sequence ofρ-mixing random variables withEYi=0,0&lt;EYi2&lt;∞,∑i=1∞ρ(2i)&lt;∞. The results obtained generalize the results of Liang et al. (2004) toρ-mixing sequences.

This book aims to introduce modern asymptotic theory to students and practitioners of econometrics. It falls broadly into two parts. The first provides a handbook and reference for the underlying mathematics (Part I, Chapters 1–6), statistical theory (Part II, Chapters 7–11), and stochastic process theory (Part III, Chapters 12–18). The second half provides a treatment of the main convergence theorems used in analysing the large sample behaviour of econometric estimators and tests. These are the law of large numbers (Part IV, Chapters 19–22), the central limit theorem (Part V, Chapters 23–26), and the functional central limit theorem (Part VI, Chapters 27–32). The focus in this treatment is on the nonparametric approach to time series properties, covering topics such as nonstationarity, mixing, martingales, and near‐epoch dependence. While the approach is not elementary, care is taken to keep the treatment self‐contained. Proofs are provided for almost all the results.

Some new asymptotic theory for least squares series: Pointwise and uniform results

In this article, we study the hyperbolic Anderson model driven by a space-time colored Gaussian homogeneous noise with spatial dimension d=1,2. Under mild assumptions, we provide L^p-estimates of the iterated Malliavin derivative of the solution in terms of the fundamental solution of the wave solution. To achieve this goal, we rely heavily on the Wiener chaos expansion of the solution. Our first application are quantitative central limit theorems for spatial averages of the solution to the hyperbolic Anderson model, where the rates of convergence are described by the total variation distance. These quantitative results have been elusive so far due to the temporal correlation of the noise blocking us from using the Itô calculus. A novel ingredient to overcome this difficulty is the second-order Gaussian Poincaré inequality coupled with the application of the aforementioned L^p-estimates of the first two Malliavin derivatives. Besides, we provide the corresponding functional central limit theorems. As a second application, we establish the absolute continuity of the law for the hyperbolic Anderson model. The L^p-estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continuity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by [2].

Guided by the self-interaction mechanisms introduced in Benaim et al. [2] and in [5], we present a more general definition of self-interacting Markov chains (SIMCs) (than in Del Moral and Miclo [5] and Benaim et al. [2]). We then establish, for particular self-interaction mechanisms, a stability theorem with error estimation, two central limit theorems, two functional central limit theorems, and the large deviation principle.

A functional central limit theorem is obtained for a stationary linear process of the form Xt = P1=0 aj†tij, where f†tg is a strictly stationary associated sequence of random variables with E†t = 0; E(† 2) < 1 and fajg is a sequence of real numbers with P1 j=0 jajj < 1: A central limit theorem for a stationary linear pro- cess generated by stationary associated processes is also discussed.

In this paper we develop and validate a procedure for testing against a shift in mean in the observa-tions and hidden state sequence of state space models with Gaussian noise. State space models are popular for modelling stochastic networks as they allow to take into account that observations of the true state of a sys-tem may be corrupted by measurement noise (usually, a Gaussian noise process is assumed). Although state space models are very general, they are still relatively tractable in that the true system state can be estimated efficiently by a recursive procedure known as Kalman filtering. State space models can be regarded as a special type of hidden Markov model. As such, they are a flexible modelling tool that has been found useful, for example, for modelling road networks (Stathopoulos and Karlaftis, 2003) to account for uncertainty in the measurement of travel times. For instance, we may assume that travel times have to be estimated from flow and occupancy data. An increase in the unobserved mean travel time can be caused by traffic congestion; a shift in the mean value of the observations on the other hand could indicate a bias of the sensors. State space models can also be used to model communication networks: Suppose that the current state of a channel (e.g. measured by the probability of packet loss) is not observed directly, but has to be inferred from the received package flow. A change in the mean value of the hidden state sequence, or a change in the mean value of the received package flow can deteriorate the performance of the network if it remains unrecognised. This motivates us to investigate procedures for testing against a shift in mean in the observations and hidden state sequence of state space models. The objective is to detect a change as quickly as possible while keeping the ratio of false alarms at a pre-specified low level. Since the observations are generally not independent, in (Basseville and Nikiforov, 1993) a cumulative sum (CUSUM) procedure is applied to the (independent) sequence of innovations, which is obtained as a byproduct from Kalman filter estimation of the hidden states. That is, a log-likelihood ratio (LLR) test statistic is used and an alarm is raised as soon as this test statistic exceeds a certain threshold that is assumed to be given. Change point detection for state space models has also been considered in (Lai and Shan, 1999) for the case where the size of the mean shift is unknown, in which case a generalized LLR test can be applied. In this paper we tackle the question of how the threshold of the sequential LLR test can be chosen when the shift size is assumed to be known. In practice, the latter assumption can be dealt with by realizing that typically there will be a minimum change size that is of interest from an engineering perspective, and that can thus be used as input for the model. Based on this assumption we can identify the appropriate level of the threshold based on approximations of the false alarm probability - essentially the probability that a random walk process exceeds a given threshold on an interval. A persistent change in the mean value of the observations results in a dynamic change in the mean value of the innovations, which are therefore not identically distributed after the change point. However, it follows from the stability properties of the Kalman filter that under weak conditions the magnitude of the shift converges to a constant. This allows for large-deviations (LD) approximations as well as approximations motivated by a functional central limit theorem (CLT). LD approximations to the false alarm probability have been considered in (Bucklew, 1985; Ellens et al., 2013; Kuhn et al., 2014) for testing i.i.d. and vector autoregressive moving average (VARMA) models. CLT approximations were motivated, for example, in Siegmund (1985). We compare the numerical performance of the tests under both types of limiting regimes with respect to the false alarm probability and the detection delay.

Summary form only given, as follows. Certain asymptotic properties of supervised learning algorithms are developed. It is noted that there is a strong resemblance between the supervised learning and adaptive filtering problems. Almost sure convergence is established. The usual IID assumption on the data is weakened. The limiting distribution of a normalized sequence is obtained. In fact, a functional central limit theorem is derived. Such asymptotic normality is then utilized to design a practical stopping rule, which is based on the construction of ellipsoidal confidence regions. To ensure the boundedness of the iterates in the learning algorithms, several truncation or projection methods are described and the convergence is discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A central limit theorem for functions of stationary max-stable random fields on [formula omitted