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.     

Laws of the iterated logarithm for partial sum processes indexed by functions

We establish a bounded and a compact law of the iterated logarithm for partial sum processes indexed by classes of functions. We assume a growth condition on the metric entropy under bracketing. Examples show that our results are sharp. As a corollary we obtain new results for weighted sums of independent identically distributed random variables.     

Rates of growth and sample moduli for weighted empirical processes indexed by sets

Summary Probability inequalities are obtained for the supremum of a weighted empirical process indexed by a Vapnik-ervonenkis class C of sets. These inequalities are particularly useful under the assumption P({CC:P(C)<t})»0 as t»0. They are used to obtain almost sure bounds on the rate of growth of the process as the sample size approaches infinity, to find an asymptotic sample modulus for the unweighted empirical process, and to study the ratio P _n/P of the empirical measure to the actual measure.Research supported under an NSF Postdoctoral Fellowship grant No. MCS 83-11686, and in part by NSF grant No. DMS-8301807     

Invariance principles for sums of Banach space valued random elements and empirical processes

Summary Almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable Banach space. It is then shown that empirical processes readily fit into this general framework. Thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes.Both authors were partially supported by NSF grants. This work was done while the second author was visiting the M.I.T. Mathematics Department     

Invariance principles for tempered fractionally integrated processes

We discuss invariance principles for autoregressive tempered fractionally integrated moving averages in $\alpha$-stable $(1<\alpha \le 2)$ i.i.d. innovations and related tempered linear processes with vanishing tempering parameter $\underset{N\to \infty }{lim}\lambda ∕N={\lambda }_1$. We show that the limit of the partial sums process takes a different form in the weakly tempered (${\lambda }_1=0$), strongly tempered (${\lambda }_1=\infty$), and moderately tempered ($0<{\lambda }_1<\infty$) cases. These results are used to derive the limit distribution of the ordinary least squares estimate of AR(1) unit root with weakly, strongly, and moderately tempered moving average errors.     

STRONG LAWS FOR α-MIXING SEQUENCE PROCESSES INDEXED BY SETS

ＳＴＲＯＮＧＬＡＷＳＦＯＲα－ＭＩＸＩＮＧＳＥＱＵＥＮＣＥＰＲＯＣＥＳＳＥＳＩＮＤＥＸＥＤＢＹＳＥＴＳ￥ＸＵＢＩＮＧＡｂｓｔｒａｃｔ：ＬｅｔＪ＝｛１，２，．．．｝ｄａｎｄｌｅｔ｛Ｘｊ，ｊ∈Ｊ｝ｂｅａｎａ－ｍｉｘｉｎｇｓｅｑｕｅｎｃｅｗｈｉｃｈｉｓｎｏ...     

Gaussian approximation of local empirical processes indexed by functions

Summary. An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated. Received: 11 January 1995/In revised form: 12 July 1996     

Functional Erdös-Rényi laws for processes indexed by sets

We consider a process X(A) indexed by some sets A∈A, where A is a collection of sets. We prove a functional form of the Erdös-Rényi laws. The result may be specialized to get back and sometimes improve previous versions of the classical Erdös-Rényi laws.     

Asymptotic behaviors for partial sum processes of a Gaussian sequence

We establish some large increment results for partial sum processes of a dependent stationary Gaussian sequence via estimating upper bounds of large deviation probabilities on suprema of the Gaussian sequence.     

Invariance principles for Gaussian sequences

An almost sure invariance principle is proved for stationary Gaussian sequences whose covariances r(n) satisfy r(n) = O (n ^–1–)for some >0.     

Clustering behavior of finite variance partial sum processes

Summary Clustering rates in Strassen's functional law of the iterated logarithm are determined for finite variance partial sum processes in one dimension. A general characterization of these rates, similar to one recently obtained for onedimensional Brownian motion, shows that relatively mild moment conditions on a partial sum process lead to high order clustering rates at certain points of the Strassen set.Supported in part by NSF Grant DMS-92-07248     

Invariance principles for von Mises and U-statistics

Summary The almost sure approximation of von Mises-statistics and U-statistics by appropriate stochastic integrals with respect to Kiefer processes is obtained. In general these integrals are non-Gaussian processes. As applications we get almost sure versions for the estimator of the variance and for the ²-test of goodness of fit.This work was done while the last author was a visiting professor at the Institut für Mathematische Stochastik at the University of Göttingen during the Spring of 1982. He thanks the Institut and its members for their hospitality     

A kriging procedure for processes indexed by graphs

We provide a new kriging procedure of processes on graphs. Based on the construction of Gaussian random processes indexed by graphs, we extend to this framework the usual linear prediction method for spatial random fields, known as kriging. We provide the expression of the estimator of such a random field at unobserved locations as well as a control for the prediction error.     

Stochastic processes indexed by hypergroups II

Further results on weakly stationary processes indexed by hypergroups are presented. The concept of translation operators is developed; processes on orbit spaces and double coset spaces are constructed. It is shown that every weakly stationary process indexed by a hypergroupK with centerC contains a maximalK//C-weakly stationary component. New examples forK-weakly stationary processes are continuous estimates of the mean of a weakly stationary process, isotropic random fields, andK-oscillations.     

Some limit theorems for the empirical process indexed by functions

Summary Let,n1, be a sequence of classes of real-valued measurable functions defined on a probability space (S,,P). Under weak metric entropy conditions on,n1, and under growth conditions on we show that there are non-zero numerical constantsC ₁ andC ₂ such that where (n) is a non-decreasing function ofn related to the metric entropy of. A few applications of this general result are considered: we obtain a.s. rates of uniform convergence for the empirical process indexed by intervals as well as a.s. rates of uniform convergence for the empirical characteristic function over expanding intervals.Portions of this article were presented during the conference on Mathematical Stochastics (February 19–25, 1984) at Oberwolfach, West Germany     

Empirical processes indexed by smooth functions

In this paper we derive a general invariance principle for empirical processes indexed by smooth functions. The method is applied to prove bounds for the convergence of the empirical distributions which might be useful to verify asymptotic normality of smooth statistical functionals. As one further application we get the convergence of the so-called empirical characteristic function process.