On the Glivenko–Cantelli theorem for generalized empirical processes based on strong mixing sequences

Given \s{X_i, i 1\s} as non-stationary strong mixing (n.s.s.m.) sequence of random variables (r.v.'s) let, for 1 i n and some γ ε [0, 1],

F₁(x)=γP(X_i<x)+(1-γ)P(X_ix)

and

I_i(x)=γI(X_i<x)+(1-γ)I(X_ix)

. For any real sequence \s{C_i\s} satisfying certain conditions, let

.

In this paper an exponential type of bound for P(D_n ), for any >0, and a rate for the almost sure convergence of D_n are obtained under strong mixing. These results generalize those of Singh (1975) for the independent and non-identically distributed sequence of r.v.'s to the case of strong mixing.     

Weak convergence of the sequential empirical processes of residuals in TAR models

This paper studies the weak convergence of the sequential empirical process K n of the residuals in the threshold autoregressive(TAR)model of order p.Under some mild conditions,it is shown that K n converges weakly to a Kiefer process plus a random variable which converges to a multivariate normal.This differs from that given by Bai(1994)for a stationary autoregressive and moving average(ARMA)model.     

Large deviations for empirical measures of switching diffusion processes with small parameters

We consider the asymptotic property of the diffusion processes with Markovian switching. For a general case, we prove a large deviation principle for empirical measures of switching diffusion processes with small parameters.     

Some properties of bivariate empirical hazard processes under random censoring

In Campbell (1982, IMS Lecture Notes—Monograph Series Vol. 2, pp. 243–256, IMS, Hayward, CA) and Campbell and Földes (1982, Proceedings, Internat. Colloq. Nonparametric Statist. Inform., 1980, North-Holland, New York) some asymptotic properties of bivariate empirical hazard processes under random censoring are given. Taking the representation of the empirical hazard process for bivariate randomly censored samples in Campbell, op. cit., as a starting point and restricting attention to strong properties, we obtain a speed of strong convergence for the weighted bivariate empirical hazard processes as well as a speed of strong uniform convergence for bivariate hazard rate estimators. Our approach is based on a local fluctuation inequality for the bivariate hazard process and differs from the martingale methods quite often used in the univariate case.     

On the convergence of sequences of stationary jump Markov processes

This paper presents two main results: first, a Liapunov type criterion for the existence of a stationary probability distribution for a jump Markov process; second, a Liapunov type criterion for existence and tightness of stationary probability distributions for a sequence of jump Markov processes. If the corresponding semigroups T_N(t) converge, under suitable hypotheses on the limit semigroup, this last result yields the weak convergence of the sequence of stationary processes (T_N(t), π_N) to the stationary limit one.     

A test for independence of two stationary infinite order autoregressive processes

This paper considers the independence test for two stationary infinite order autoregressive processes. For a test, we follow the empirical process method and construct the Cramér-von Mises type test statistics based on the least squares residuals. It is shown that the proposed test statistics behave asymptotically the same as those based on true errors. Simulation results are provided for illustration.     

Multiscale Young measures in homogenization of continuous stationary processes in compact spaces and applications

We introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces. The latter are functions f○T:Rn×Q→Q with f○T(x,ω)=f(T(x)ω) where Q is a compact (Hausdorff topological) space, f∈C(Q) and T(x):Q→Q, x∈Rn, is an n-dimensional continuous dynamical system endowed with an invariant Radon probability measure μ. It can be easily shown that for almost all ω∈Q the realization f(T(x)ω) belongs to an algebra with mean value, that is, an algebra of functions in BUC(Rn) containing all translates of its elements and such that each of its elements possesses a mean value. This notion was introduced by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators, Mat. Zametki 33 (1983) 571-582, English transl. in Math. Notes 33 (1983) 294-300]. We then establish the existence of multiscale Young measures in the setting of algebras with mean value, where the compactifications of Rn provided by such algebras plays an important role. These parametrized measures are useful in connection with the existence of correctors in homogenization problems. We apply this framework to the homogenization of a porous medium type equation in Rn with a stationary continuous process as a stiff oscillatory external source. This application seems to be new even in the classical context of periodic homogenization.     

The limit theorems for maxima of stationary Gaussian processes with random Index

Let {X(t), t ≥ 0} be a standard(zero-mean, unit-variance) stationary Gaussian process with correlation function r(·) and continuous sample paths. In this paper, we consider the maxima M(T) = max{X(t), t∈ [0, T ]} with random index TT, where TT /T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M(TT) exists under some additional conditions related to the correlation function r(·).     

On tails of stationary measures on a class of solvable groups

Let G be a subgroup of GL(R,d) and let (Qn,Mn) be a sequence of i.i.d. random variables with values in Rd?G and law μ. Under some natural conditions there exists a unique stationary measure ν on Rd of the process Xn=MnXn−1+Qn. Its tail properties, i.e. behavior of as t tends to infinity, were described some over thirty years ago by H. Kesten, whose results were recently improved by B. de Saporta, Y. Guivarc'h and E. Le Page. In the present paper we study the tail of ν in the situation when the group G₀ is Abelian and Rd is replaced by a more general nilpotent Lie group N. Thus the tail behavior of ν is described for a class of solvable groups of type NA, i.e. being semi-direct extension of a simply connected nilpotent Lie group N by an Abelian group isomorphic to Rd. Then, due to A. Raugi, (N,ν) can be interpreted as the Poisson boundary of (NA,μ).     

Approximation of Gaussian measures generated by stationary processes. I

We consider local approximation of Gaussian measures generated by stationary processes, which are smooth functions of a parameter.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 79, pp. 44–66, 1978.     

The Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes

The equivalence of ergodicity and weak mixing for general infinitely divisible processes is proven. Using this result and [9], simple conditions for ergodicity of infinitely divisible processes are derived. The notion of codifference for infinitely divisible processes is investigated, it plays the crucial role in the proofs but it may be also of independent interest.     

Weak convergence of empirical and quantile processes in sup-norm metrics via kmt-constructions

There has been much interest recently in the specially constructed empirical processes of Komlós, Major and Tusnády [2]; as one would guess, much of the application has come from the Hungarian school.In this note we contribute to the unifying effect this profound work has had by showing how the major theorem of O'Reilly [4] follows in rather elementary fashion from this powerful construction. We also take this opportunity to restate O'Reilly's criterion in an elementary form that is far more intelligible.     

Invariance principles and Gaussian approximation for strictly stationary processes

We show that in any aperiodic and ergodic dynamical system there exists a square integrable process the partial sums of which can be closely approximated by the partial sums of Gaussian i.i.d. random variables. For both weak and strong invariance principles hold.

     

Variation analysis of uncertain stationary independent increment processes

A stationary independent increment process is an uncertain process with stationary and independent increments. This paper aims to calculate the variance of stationary independent increment processes, and gains that, for each fixed time, the variance is a constant multiplying the square of time. Based on this result, it is proved that the total variation of stationary independent increment process with finite variance is bounded almost surely. Besides, the quadratic variation of stationary independent increment process with finite variance is 0 almost surely and in mean.     

The central limit theorem for weighted empirical processes indexed by sets

Sufficient conditions are found for the weak convergence of a weighted empirical process {(ν_n(C)/q(P(C))) 1 _{[P(C) λn]}: C }, indexed by a class of sets and weighted by a function q of the size of each set. We find those functions q which allow weak convergence to a sample-continuous Gaussian process, and, given q, determine the fastest rate at which one may allow λ_n → 0.     

Testing for independence by the empirical characteristic function

Tests of total independence of d (≥2) random variables are proposed using the empirical characteristic function. The approach is parallel to that of Hoeffding, Blum, Kiefer, and Rosenblatt.     

Stable sub-Gaussian models constructed by Poisson processes

In this paper, we first prove that one-parameter standard α-stable sub-Gaussian processes can be approximated by processes constructed by integrals based on the Poisson process with random intensity. Then we extend this result to the two-parameter processes. At last, we consider the approximation of the subordinated fractional Brownian motion.     

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.     

The low of the iterated logarithm for empirical processes under absolute regularity

The compact law of the iterated logarithm for empirical processes whose underlying sequence satisfies a -mixing condition is considered. In particular, we show a compact law of the iterated logarithm for VC subgraph classes of functions, for classes of functions which satisfy the bracketing condition in Doukhanet al. ⁽⁶⁾ and for some classes of smooth functions.Research partially supported by NSF Grant DMS-93-02583.     

Bounds for the variance of certain stationary point processes

For the variance of stationary renewal and alternating renewal processes N_n(·) the paper establishes upper and lower bounds of the form

$\text{?B}{}_1\text{?varN}{}_8\text{(0,x–Aλx?B}{}_2\text{(0<x<∞)}$

, where λ=EN₈(0,1), with constants A, B₁ and B₂ that depend on the first three moments of the interval distributions for the processes concerned. These results are consistent with the value of the constant A for a general stationary point process suggested by Cox in 1963 [1].