On the mixing coefficients of piecewise monotonic maps

We investigate the mixing coefficients of interval maps satisfying Rychlik's conditions. A mixing Lasota-Yorke map is reverse ϕ-mixing. If its invariant density is uniformly bounded away from 0, it is ϕ-mixing iff all images of all orders are big, in which case it is ψ-mixing. Among β-transformations, non-ϕ-mixing is generic. In this sense, the asymmetry of ϕ-mixing is natural.     

Convergence rates in the strong laws of asymptotically negatively associated random fields

In this paper, a notion of negative side p-mixing (p -mixing) which can be regardedas asymptotic negative association is defined, and some Rosenthal type inequalities for p -mix-ing random fields are established. The complete convergence and almost sure summability onthe convergence rates with respect to the strong law of large numbers are also discussed for p--mixing random fields. The results obtained extend those for negatively associated sequences andp“ -mixing random fields.     

Change-point Estimation of a Mean Shift in Moving-average Processes Under Dependence Assumptions

In this paper we discuss the least-square estimator of the unknown change point in a mean shift for moving-average processes of ALNQD sequence. The consistency and the rate of convergence for the estimated change point are established. The asymptotic distribution for the change point estimator is obtained. The results are also true for ρ-mixing, φ-mixing, α-mixing sequences under suitable conditions. These results extend those of Bai, who studied the mean shift point of a linear process of i.i.d, variables, and the condition ∑j=0^∞j｜aj｜ 〈 ∞ in Bai is weakened to ∑j=0^∞｜aj｜〈∞.     

On the global central limit theorem for ϕ-mixing random variables

The estimate of the remainder term is obtained in the global central limit theorem for π-mixing r.v.s. As a consequence of Theorem 1 the convergence rate of absolute moments for sums of π-mixing r.v.s. to corresponding absolute moments of the normal r.v. is found. Published in Lietuvos Matematikos Rinkinys, Vol. 35, No. 2, pp. 233–247, April–June, 1995.     

Analysis of rounded data from dependent sequences

Observations on continuous populations are often rounded when recorded due to the precision of the recording mechanism. However, classical statistical approaches have ignored the effect caused by the rounding errors. When the observations are independent and identically distributed, the exact maximum likelihood estimation (MLE) can be employed. However, if rounded data are from a dependent structure, the MLE of the parameters is difficult to calculate since the integral involved in the likelihood equation is intractable. This paper presents and examines a new approach to the parameter estimation, named as “short, overlapping series” (SOS), to deal with the α-mixing models in presence of rounding errors. We will establish the asymptotic properties of the SOS estimators when the innovations are normally distributed. Comparisons of this new approach with other existing techniques in the literature are also made by simulation with samples of moderate sizes.     

A note on the Bahadur representation of sample quantiles for α-mixing random variables

By some moment inequalities for α-mixing random variables, we prove the Bahadur representation of sample quantiles under very weak α-mixing coefficients. As application, the uniformly asymptotic normality is derived, the rate of which is near to n ^−1/6 under the given conditions.     

一类独立随机场的强定律的收敛率

§ 1　IntroductionDefinition1 .[1 ] A field{ Xi,i∈Nd} is called negatively associated(NA) if for every pair ofdisjoint subsets T1 ,T2 of Nd,Cov(f1 (Xi,i∈ T1 ) ,f2 (Xj,j∈ T2 ) )≤ 0 ,whenever f1 and f2 are coordinatewise increasing.Definition2 .[1 ] A field{ Xi,i∈Nd} is calledρ* -mixing ifρ* (s) =sup{ (ρ(S,T) ;S,T N,dist(S,T)≥ s}→ 0 (s→∞ ) ,whereρ(S,T) =sup{ |E(f -Ef) (g -Eg) |/‖ f -Ef‖2 ‖ g -Eg‖2 ,f∈ L2 (σ(S) ) ,g∈ L2 (σ(T) ) } .Definition 3.[1 ] A field { Xi…     

Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k-r), 0 < r ≤1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(l+r)/logn)-1/2) or O((nr/(1+r)/ log n)-3/4) and that the Central Limit Theorem does not always hold for the empirical processes.``     

Estimating the Distribution Function of a Stationary Process Involving Measurement Errors

A nonparametric estimation of a distribution function is considered when observations contain measurement errors. A method is developed to establish asymptotic normality results for a deconvoluting kernel-type estimator for ρ-mixing stochastic processes corrupted by some noise process. It is shown that the asymptotic distribution depends on the smoothness of the noise distributions, which are characterized as either ordinary smooth or super smooth. Also, the kind of dependence of the noise process is crucial to the form of the asymptotic variance. This revised version was published online in June 2006 with corrections to the Cover Date.     

φ-混合样本下有限个分位数估计的联合渐近分布

本文研究了φ-混合样本下总体的有限个分位数核估计的渐近性质.利用分块技术证明了φ-混合样本下总体的有限个分位数核估计的联合渐近分布为多元正态分布,推广了文献[16]的相关结果.     

A central limit theorem for stationary random fields

Summary. We prove a central limit theorem for strictly stationary random fields under a projective assumption. Our criterion is similar to projective criteria for stationary sequences derived from Gordin's theorem about approximating martingales. However our approach is completely different, for we establish our result by adapting Lindeberg's method. The criterion that it provides is weaker than martingale-type conditions, and moreover we obtain as a straightforward consequence, central limit theorems for α-mixing or φ-mixing random fields. Received: 19 February 1997 / In revised form: 2 September 1997     

Laws of the iterated logarithm and an almost sure invariance principle for mixing <Emphasis Type="Italic">B</Emphasis>-valued random variables and autoregressive processes

Under optimal moment conditions, we prove the compact law of the iterated logarithm and the almost sure invariance principle for ψ-mixing random variables with values in type 2 Banach spaces. These results, together with the bounded law of the iterated logarithm proved earlier by author, allow us to prove the same kind of results for the Banach space valued autoregressive processes with ψ-mixing innovations. The results for autoregressive processes can be considered as asymptotic properties of the estimator of mean.     

On the rosenthal inequality for mixing fields

A proof of the Rosenthal inequality for α-mixing random fields is given. The statements and proofs are modifications of the corresponding results obtained by Doukhan and Utev.     

Dependence and the dimensionality reduction principle

Stone’s dimensionality reduction principle has been confirmed on several occasions for independent observations. When dependence is expressed with ϕ-mixing, a minimum distance estimate is proposed for a smooth projection pursuit regression-type function θ∈Я, that is either additive or multiplicative, in the presence of or without interactions. Upper bounds on theL ₁-risk and theL ₁-error of are obtained, under restrictions on the order of decay of the mixing coefficient. The bounds show explicitly the addive effect of ϕ-mixing on the error, and confirm the dimensionality reduction principle.     

On L 1 Bounds for Asymptotic Normality of Some Weakly Dependent Random Variables

In the present paper, we consider L ₁ bounds for asymptotic normality for the sequence of r.v.’s X ₁,X ₂,… (not necessarily stationary) satisfying the ψ-mixing condition. The L ₁ bounds have been obtained in terms of Lyapunov fractions which, in a particular case, under finiteness of the third moments of summands and the finiteness of ∑ _r≥1 r ² ψ(r), are of order O(n ^−1/2), where the function ψ participates in the definition of the ψ-mixing condition.      

Regularized least square regression with dependent samples

In this paper we study the learning performance of regularized least square regression with α-mixing and ϕ-mixing inputs. The capacity independent error bounds and learning rates are derived by means of an integral operator technique. Even for independent samples our learning rates improve those in the literature. The results are sharp in the sense that when the mixing conditions are strong enough the rates are shown to be close to or the same as those for learning with independent samples. They also reveal interesting phenomena of learning with dependent samples: (i) dependent samples contain less information and lead to worse error bounds than independent samples; (ii) the influence of the dependence between samples to the learning process decreases as the smoothness of the target function increases.     

Self-crossing Points of a Line Segment Process

This paper is devoted to planar stationary line segment processes. The segments are assumed to be independent, identically distributed, and independent of the locations (reference points). We consider a point process formed by self-crossing points between the line segments. Its asymptotic variance is explicitly expressed for Poisson segment processes. The main result of the paper is the central limit theorem for the number of intersection points in expanding rectangular sampling window. It holds not only for Poisson processes of reference points but also for stationary point processes satisfying certain conditions on absolute regularity (β-mixing) coefficients. The proof is based on the central limit theorem for β-mixing random fields. Approximate confidence intervals for the intensity of intersections can be constructed.     

LIL behavior for weakly dependent random variables in Banach spaces

Given a sequence of identically distributed ψ-mixing random variables {X _n ; n ≧ 1} with values in a type 2 Banach space B, under certain conditions, the law of the iterated logarithm for this sequence is obtained without second moment.     

On the discounted global CLT for some weakly dependent random variables

In this paper, we consider L ₁ upper bounds in the global central limit theorem for the sequence of r.v.’s (not necessarily stationary) satisfying the ψ-mixing condition. In a particular case, under the finiteness of the third absolute moments of summands A _i and that of the series ∑ _r⩾1 r ² φ(r), we obtain bounds of order O(n ^−1/2) for Δ _n1:= ∫ _−∞ ^∞ |ℙ{A ₁ + ⋯ + A _n < x} − Φ(x)|dx, where is the standard normal distribution function, and ψ is the function participating in the definition of the ψ-mixing condition. Moreover, we apply the obtained results to get the convergence rate in the so-called discounted global CLT for a sequence of r.v.’s, satisfying the ψ-mixing condition. The bounds obtained provide convergence rates in the discounted global CLT of the same order as in the case of i.i.d. summands with a finite third absolute moment, i.e., of order O((1 − υ)^1/2), where υ is a discount factor, 0 < υ < 1. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 584–597, October–December, 2006.     

A Note on the Strong Approximation of the Smoothed Empirical Process of α-mixing Sequences

A strong approximation of the smoothed empirical process of strictly stationary α-mixing random variables by a sequence of iid Gaussian processes will be studied. Here, the smoothing is done via kernel density estimators. No assumptions are made on the support of the kernel; in fact, our main results are stated for kernels with possibly an infinite support. Received June 2003; Accepted February 2004.