An almost sure central limit theorem for self-normalized partial sums of weakly dependent random variables

Abstract

We give here an almost sure central limit theorem for self-normalized partial sums of a strictly stationary φ-mixing sequences which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Our result substantially extend a result on the almost sure central limit theorem previously obtained by Huang and Pang (2010).     

On functional central limit theorems for dependent,heterogeneous arrays with applications to tail index and tail dependence estimation

We establish invariance principles for a large class of dependent, heterogeneous arrays. The theory equally covers conventional arrays, and inherently degenerate tail arrays popularly encountered in the extreme value theory literature including sample means and covariances of tail events and exceedances. For tail arrays we trim dependence assumptions down to a minimum leaving non-extremes and joint distributions unrestricted, covering geometrically ergodic, mixing, and mixingale processes, in particular linear and nonlinear distributed lags with long or short memory, linear and nonlinear GARCH, and stochastic volatility.     

Central limit theorems for LS estimators in the EV regression model with dependent measurements

In this paper, we consider the simple linear errors-in-variables (EV) regression models: η_i=θ+βx_i+ε_i,ξ_i=x_i+δ_i,1≤i≤n, where θ,β,x₁,x₂,… are unknown constants (parameters), (ε₁,δ₁),(ε₂,δ₂),… are errors and ξ_i,η_i,i=1,2,… are observable. The asymptotic normality for the least square (LS) estimators of the unknown parameters β and θ in the model are established under the assumptions that the errors are m-dependent, martingale differences, ?-mixing, ρ-mixing and α-mixing.     

Root-n-consistent semiparametric estimation of partially linear models for weakly dependent observations

We consider estimation of the linear part in a partially linear model for absolutely regular observations. The estimator using random weights are proposed and the asymptotic normality of the estimator is established without compact support assumption.     

On functional central limit theorems of Bayesian nonparametric priors

A general approach to derive the weak convergence, when centered and rescaled, of certain Bayesian nonparametric priors is proposed. This method may be applied to a wide range of processes including, for instance, nondecreasing nonnegative pure jump Lévy processes and normalized nondecreasing nonnegative pure jump Lévy processes with known finite dimensional distributions. Examples clarifying this approach involve the beta process in latent feature models and the Dirichlet process.     

Quantitative central limit theorems for Mexican needlet coefficients on circular Poisson fields

The aim of this paper is to establish rates of convergence to Gaussianity for wavelet coefficients on circular Poisson random fields. This result is established by using the Stein–Malliavin techniques introduced by Peccati and Zheng (Electron J Probab 15(48):1487–1527, 2010) and the concentration properties of so-called Mexican needlets on the circle.     

Central limit theorems for smoothed extreme value estimates of Poisson point processes boundaries

In this paper, we give sufficient conditions to establish central limit theorems for boundary estimates of Poisson point processes. The considered estimates are obtained by smoothing some bias-corrected extreme values of the point process. We show how the smoothing leads to Gaussian asymptotic distributions and therefore pointwise confidence intervals. Some new unidimensional and multidimensional examples are provided.     

A goodness-of-fit test for multivariate multiparameter copulas based on multiplier central limit theorems

Recent large scale simulations indicate that a powerful goodness-of-fit test for copulas can be obtained from the process comparing the empirical copula with a parametric estimate of the copula derived under the null hypothesis. A first way to compute approximate p-values for statistics derived from this process consists of using the parametric bootstrap procedure recently thoroughly revisited by Genest and Rémillard. Because it heavily relies on random number generation and estimation, the resulting goodness-of-fit test has a very high computational cost that can be regarded as an obstacle to its application as the sample size increases. An alternative approach proposed by the authors consists of using a multiplier procedure. The study of the finite-sample performance of the multiplier version of the goodness-of-fit test for bivariate one-parameter copulas showed that it provides a valid alternative to the parametric bootstrap-based test while being orders of magnitude faster. The aim of this work is to extend the multiplier approach to multivariate multiparameter copulas and study the finite-sample performance of the resulting test. Particular emphasis is put on elliptical copulas such as the normal and the t as these are flexible models in a multivariate setting. The implementation of the procedure for the latter copulas proves challenging and requires the extension of the Plackett formula for the t distribution to arbitrary dimension. Extensive Monte Carlo experiments, which could be carried out only because of the good computational properties of the multiplier approach, confirm in the multivariate multiparameter context the satisfactory behavior of the goodness-of-fit test.     

A uniform central limit theorem for neural network-based autoregressive processes with applications to change-point analysis

We consider an autoregressive process with a nonlinear regression function that is modelled by a feedforward neural network. First, we derive a uniform central limit theorem which is useful in the context of change-point analysis. Then, we propose a test for a change in the autoregression function which – by the uniform central limit theorem – has asymptotic power one for a large class of alternatives including local alternatives not restricted to the correctly specified model.     

Limit theorems for dependent Bernoulli variables with statistical inference

We consider a class of dependent Bernoulli variables where the conditional success probability is a linear combination of the last few trials and the original success probability. We obtain its limit theorems including the strong law of large numbers, weak invariance principle, and law of the iterated logarithm. We also derive some statistical inference results which make the model applicable. Simulation results are exhibited as well to show that with small sample size the convergence rate is satisfying and the proposed estimators behave well.     

Some limit theorems for uniform and exponential spacings

The asymptotic distributions of spacing statistics based on exponential and uniform spacings are studied, and the corresponding rates of convergence are also given.     

Large deviation local limit theorems for ratio statistics

Let {T_n, n ≥ 1} be an arbitrary sequence of nonlattice random variables and let {S_n, n ≥ 1} be another sequence of positive random variables. Assume that the sequences are independent. In this paper we obtain asymptotic expression for the density function of the ratio statistic R_n = T_n/S_n based on simple conditions on the moment generating functions of T_n and S_n. When S_n = re, our main result reduces to that of Chaganty and Sethura-man[Ann. Probab. 13(1985):97-114]. We also obtain analogous results when T_n and S_n are both lattice random variables. We call our theorems large deviation local limit theorems for R_n, since the conditions of our theorems imply that R_n → c in probability for some constant c. We present some examples to illustrate our theorems.     

Stability theorems for maxima of dependent sequences

Three stability theorems due to Gnedenko (1943), Barndorff-Nielsen (1963), and Tomkins (1986) are extended to the sample maxima of identically distributed ?-mixing and m-dependent sequences.     

Dependent Lindeberg central limit theorem for the fidis of empirical processes of cluster functionals

Drees H. and Rootzén H. [Limit theorems for empirical processes of cluster functionals (EPCF). Ann Stat. 2010;38(4):2145–2186] have proven central limit theorems (CLTs) for EPCF built from β-mixing processes. However, this family of β-mixing processes is quite restrictive. We expand some of those results, for the finite-dimensional marginal distributions (fidis), to a more general dependent processes family, known as weakly dependent processes in the sense of Doukhan P. and Louhichi S. [A new weak dependence condition and applications to moment inequalities. Stoch. Proc. Appl. 1999;84:313–342]. In this context, the CLT for the fidis of EPCF is sufficient in some applications. For instance, we prove the convergence without mixing conditions of the extremogram estimator, including a small example with simulation of the extremogram of a weakly dependent random process but nonmixing, in order to confirm the efficacy of our result.     

Analysis of multivariate arma processes with non-stationary innovations

The paper considers vector ARMA processes with nonstationary innovations. It is suggested that this class of models provide a very efficient framework for nonstationary problems. A generalization of the Yule-Walker equations relating the underlying process is obtained. Identification procedures are discussed. The associated prediction problem is solved using the Hilbert space approach.     

On some moving-average processes with exponentially distributed innovations

Summary In this paper we discusse the stationary sequence of random variables which are formed from an independent identically distributed sequence, according to the moving-average model of ordern. Some properties of the process are considered. The joint bivariate exponential distribution is given, as well as the distribution of the sum.     

Central limit theorems of range-based estimators for diffusion models

Abstract

In this article, we consider non parametric range-based estimation procedure for diffusion processes and propose a instantaneous volatility estimator. Under some weak conditions, we certify that the proposed estimator has convergence in probability. Adding some necessary conditions, we prove a central limit theorem. By inference, we reach a conclusion that, with high frequency data in hand, the proposed estimator is more precise than those pure realized instantaneous volatility ones. Numerical simulation illustrates the finite sample properties of the proposed estimator.     

Central limit theorems for four new types of U-designs

Orthogonal array (OA)-based Latin hypercube designs, also called U-designs, have been popularly adopted in designing a computer experiment. Nested U-designs, sliced U-designs, strong OA-based U-designs and correlation controlled U-designs are four types of extensions of U-designs for different applications in computer experiments. Their elaborate multi-layer structure or multi-dimensional uniformity, which makes them desirable for different applications, brings difficulty in analysing the related statistical properties. In this paper, we derive central limit theorems for these four types of designs by introducing a newly constructed discrete function. It is shown that the means of the four samples generated from these four types of designs asymptotically follow the same normal distribution. These results are useful in assessing the confidence intervals of the gross mean. Two examples are presented to illustrate the closeness of the simulated density plots to the corresponding normal distributions.