Φ-mixing Sequence Research Articles

On Ibragimov–Iosifescu conjecture for φ-mixing sequences

The aim of this paper is to give new central limit theorems and invariance principles for φ-mixing sequences of random variables that support the Ibragimov–Iosifescu conjecture. A related conjecture is formulated and a positive answer is given for the distributions that have tails regularly varying with the exponent –2.

Conditional empirical processes defined by φ - mixing sequences

For a random vector (X, Y) in R 2 let m (y/x)=P(Y≤y|X=x) be the conditional distribution function. It is assumed that (X 1 , Y 1 ), (X 2 , Y 2 ), ... form a strictly stationary Φ-mixing sequence of random vectors with the same distribution as (X, Y). Considering an estimate mn (•|x) of m(•|x) it is shown that (nan) 1/2 {mn(•|x 0 )-m (•|x 0 )} converges to a certain Gaussian process depending on x 0 , where (an) is a sequence of bandwiths

Asymptotic normality of spectral estimates

The asymptotic normality of some spectral estimates, including a functional central limit theorem for an estimate of the spectral distribution function, is proved for fourth-order stationary processes. In contrast to known results it is not assumed that all moments exist or that the process is linear. The data are allowed to be tapered. Using some recent results on the central limit theorem for stationary processes, corollaries are obtained for strong and φ-mixing sequences and linear transformations of martingale differences.

Uniform Consistency of Kernel Estimators of a Regression Function under Generalized Conditions

Abstract In this article we prove uniform consistency of kernel estimators of a multivariate regression function under various assumptions on the distribution of the data. In addition to the usual assumptions that the data are iid and that the distribution of the regressors is absolutely continuous, we consider the cases that some regressors are discrete and the data are either stationary ϕ-mixing themselves or generated by a class of functions of one-sided infinite stationary ϕ-mixing sequences. Moreover, we demonstrate the performance of the kernel estimation method under these generalized conditions by a numerical example.

The invariance principle for ϕ-mixing sequences

In this paper we investigate the invariance principle for ϕ-mixing sequences, satisfying restrictions on the variances which are a weak form of stationarity. No mixing rate is assumed. For ϕ-mixing strictly stationary sequences we give a necessary and sufficient condition for the invariance principle.

Note on multidimensional empirical processes for φ-mixing random vectors

In 1974, Sen proved weak convergence of the empirical processes (in the J 1-topology on D p [0, 1]) for a stationary φ-mixing sequence of stochastic p(≧ 1)-vectors. In this note, we show that Sen's theorem on weak convergence of the multidimensional empirical process for a stationary φ-mixing sequence of stochastic vectors remains true under a less restrictive condition on the mixing constants { φ n }, i.e., φ n = O( n −1−δ) for some δ > 0.

Asymptotic normality of H-L estimators based on dependent data

LetX 1,X 2, ... be a strictly stationary φ-mixing sequence of r.v.'s with a common continuous cdfF. Let θ be a location parameter ofF. We prove the asymptotic normality of a class of Hodges-Lehmann estimators of θ under various regularity conditions on the mixing number φ and the underlyingF. We also establish the asymptotic linearity of signed rank statistics in the parameter θ.

A Central Limit Theorem with Conditioning on the Distant Past

Serfling (1968) has considered a central limit theorem in which assumptions are made concerning the expectation of variables conditioned on their distant predecessors. Dvoretsky (1972, theorem 5.3) has continued this investigation. Serfling showed that both martingales and φ-mixing sequences satisfied his conditions, and Dvoretsky extended this to Strong mixing sequences of random variables.

Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.