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Let $(X_{n,i})_{1\le i\le n,n\in\mathbb{N}}$ be a triangular array of row-wise stationary $\mathbb{R}^d$-valued random variables. We use a "blocks method" to define clusters of extreme values: the rows of $(X_{n,i})$ are divided into $m_n$ blocks $(Y_{n,j})$, and if a block contains at least one extreme value, the block is considered to contain a cluster. The cluster starts at the first extreme value in the block and ends at the last one. The main results are uniform central limit theorems for empirical processes $Z_n(f):=\frac{1}{\sqrt {nv_n}}\sum_{j=1}^{m_n}(f(Y_{n,j})-Ef(Y_{n,j})),$ for $v_n=P\{X_{n,i}\neq0\}$ and $f$ belonging to classes of cluster functionals, that is, functions of the blocks $Y_{n,j}$ which only depend on the cluster values and which are equal to 0 if $Y_{n,j}$ does not contain a cluster. Conditions for finite-dimensional convergence include $\beta$-mixing, suitable Lindeberg conditions and convergence of covariances. To obtain full uniform convergence, we use either "bracketing entropy" or bounds on covering numbers with respect to a random semi-metric. The latter makes it possible to bring the powerful Vapnik--\v{C}ervonenkis theory to bear. Applications include multivariate tail empirical processes and empirical processes of cluster values and of order statistics in clusters. Although our main field of applications is the analysis of extreme values, the theory can be applied more generally to rare events occurring, for example, in nonparametric curve estimation.

Let $(X_{n,i})_{1\le i\le n,n\in\mathbb{N}}$ be a triangular array of row-wise stationary $\mathbb{R}^d$-valued random variables. We use a "blocks method" to define clusters of extreme values: the rows of $(X_{n,i})$ are divided into $m_n$ blocks $(Y_{n,j})$, and if a block contains at least one extreme value, the block is considered to contain a cluster. The cluster starts at the first extreme value in the block and ends at the last one. The main results are uniform central limit theorems for empirical processes $Z_n(f):=\frac{1}{\sqrt {nv_n}}\sum_{j=1}^{m_n}(f(Y_{n,j})-Ef(Y_{n,j})),$ for $v_n=P\{X_{n,i}\neq0\}$ and $f$ belonging to classes of cluster functionals, that is, functions of the blocks $Y_{n,j}$ which only depend on the cluster values and which are equal to 0 if $Y_{n,j}$ does not contain a cluster. Conditions for finite-dimensional convergence include $\beta$-mixing, suitable Lindeberg conditions and convergence of covariances. To obtain full uniform convergence, we use either "bracketing entropy" or bounds on covering numbers with respect to a random semi-metric. The latter makes it possible to bring the powerful Vapnik--\v{C}ervonenkis theory to bear. Applications include multivariate tail empirical processes and empirical processes of cluster values and of order statistics in clusters. Although our main field of applications is the analysis of extreme values, the theory can be applied more generally to rare events occurring, for example, in nonparametric curve estimation.

We propose to use the rescaled range analysis to examine the records of rapidity-dependence of multiplicities in high-energy collision processes. We probe event by event the existence of global statistical dependence in the system of produced hadrons, and demonstrate the effectiveness of the above-mentioned statistical method by applying it to the cosmic-ray data of the JACEE collaboration, and by comparing the obtained results with other experimental results for similar reactions at accelerator and collider energies. We present experimental evidence for the validity of Hurst's empirical law, and the evidence for the existence of global statistical dependence, fractal dimension, and scaling behavior in such systems of hadronic matter. None of these features is directly related to the basis of the conventional physical picture. Hence, it is not clear whether (and if yes, how and why) these striking empirical regularities can be understood in terms of the conventional theory.

We propose to use the rescaled range analysis to examine the records of rapidity-dependence of multiplicities in high-energy collision processes. We probe event by event the existence of global statistical dependence in the system of produced hadrons, and demonstrate the effectiveness of the above-mentioned statistical method by applying it to the cosmic-ray data of the JACEE collaboration, and by comparing the obtained results with other experimental results for similar reactions at accelerator and collider energies. We present experimental evidence for the validity of Hurst's empirical law, and the evidence for the existence of global statistical dependence, fractal dimension, and scaling behavior in such systems of hadronic matter. None of these features is directly related to the basis of the conventional physical picture. Hence, it is not clear whether (and if yes, how and why) these striking empirical regularities can be understood in terms of the conventional theory.

We develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the supremum norm. We prove an abstract approximation theorem that is applicable to a wide variety of problems, primarily in statistics. In particular, the bound in the main approximation theorem is non-asymptotic and the theorem does not require uniform boundedness of the class of functions. The proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors, the proof of which depends on an effective use of Stein's method for normal approximation, and some new empirical process techniques. We study applications of this approximation theorem to local empirical processes and series estimation in nonparametric regression where the classes of functions change with the sample size and are not Donsker-type. Importantly, our new technique is able to prove the Gaussian approximation for the supremum type statistics under weak regularity conditions, especially concerning the bandwidth and the number of series functions, in those examples.

We consider an empirical process based upon ratio of selected pair of the non-overlapping $m$-spacings generated by independent samples of arbitrary sizes. As a main result, we show that when both samples are uniformly distributed on intervals of equal lengths, this empirical process converges to a mean-centered Brownian bridge of the form $\ \displaystyle (B\circ H_{m})_{C}(v)=B(H_{m}(v))-2(2m+1)C\ \frac{(2m-1)!}{m((m-1)!)^2}(v(1-v))^m\int_{0}^{1}B(H_{m}(s))\ud s,\ $ for $\ \displaystyle 0\leq v\leq 1,\ $ where $\ \displaystyle B(.)\ $ denotes a Brownian bridge, $\ \displaystyle H_{m},\ $ the distribution function of the Beta distribution with parameters $m$ and $m,\ $ and $C,\ $ a constant.

Bayes estimators are derived by means of the Dirichlet process hyperprior approach for general empirical Bayes problems. For any sample size, these estimators are expressed concisely as ratios of two multidimensional integrals. A numerical example on Poisson sampling is given. Keywords: Bayesian nonparametric density method; compound Poisson distribution. (Author)

We consider the convergence of empirical processes indexed by functions that depend on an estimated parameter $\eta$ and give several alternative conditions under which the ``estimated parameter'' $\eta_n$ can be replaced by its natural limit $\eta_0$ uniformly in some other indexing set $\Theta$. In particular we reconsider some examples treated by Ghoudi and Remillard [Asymptotic Methods in Probability and Statistics (1998) 171--197, Fields Inst. Commun. 44 (2004) 381--406]. We recast their examples in terms of empirical process theory, and provide an alternative general view which should be of wide applicability.

We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq 0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class $\F$ is different from the class of functions $f$ for which we have good properties of the observables $(f(X_i))_{i\geq 0}$. We introduce a new bracketing number to measure the size of the index class $\F$ which fits this setting. Our results apply to the empirical process of data $(X_i)_{i\geq 0}$ satisfying a multiple mixing condition. This includes dynamical systems and Markov chains, if the Perron-Frobenius operator or the Markov operator has a spectral gap, but also extends beyond this class, e.g.\ to ergodic torus automorphisms.

Define the scaled empirical point process on an independent and identically distributed sequence $\{Y_i: i\le n\}$ as the random point measure with masses at $a_n^{-1} Y_i$. For suitable $a_n$ we obtain the weak limit of these point processes through a novel use of a dimension-free method based on the convergence of compensators of multiparameter martingales. The method extends previous results in several directions. We obtain limits at points where the density of $Y_i$ may be zero, but has regular variation. The joint limit of the empirical process evaluated at distinct points is given by independent Poisson processes. These results also hold for multivariate $Y_i$ with little additional effort. Applications are provided both to nearest-neighbour density estimation in high dimensions, and to the asymptotic behaviour of multivariate extremes such as those arising from bivariate normal copulas.

With the recent emergence of agile software development technologies like XP, many organizations, small and large, have adopted XP or a subset of its practices. This talk describes the empirical evaluation of the XP development process in terms of industrial case studies performed at IBM, Sabre and Tekelec. These case studies involve measurement of various product and process measures to compare the XP process and also discusses the obstacles that were encountered during the case studies. This talk also provides a brief comparative introduction between XP-1 and the recent XP-2 processes. ©2005 Microsoft Corporation. All rights reserved.

In this paper we give optimal constants in Talagrand's concentration inequalities for maxima of empirical processes associated to independent and eventually nonidentically distributed random variables. Our approach is based on the entropy method introduced by Ledoux.

The distribution under alternatives of a simple linear rank statistic S sub n is governed by the following entities: a set of regression constants (series C from ml to nn), scores (series a from nl to nn) and distribution functions (series F from nl to nn). One usually assumes the scores to be generated by a known function phi.

In this paper we review important aspects of semiparametric theory and empirical processes that arise in causal inference problems. We begin with a brief introduction to the general problem of causal inference, and go on to discuss estimation and inference for causal effects under semiparametric models, which allow parts of the data-generating process to be unrestricted if they are not of particular interest (i.e., nuisance functions). These models are very useful in causal problems because the outcome process is often complex and difficult to model, and there may only be information available about the treatment process (at best). Semiparametric theory gives a framework for benchmarking efficiency and constructing estimators in such settings. In the second part of the paper we discuss empirical process theory, which provides powerful tools for understanding the asymptotic behavior of semiparametric estimators that depend on flexible nonparametric estimators of nuisance functions. These tools are crucial for incorporating machine learning and other modern methods into causal inference analyses. We conclude by examining related extensions and future directions for work in semiparametric causal inference.

We study the behavior of bivariate empirical copula process $\mathbb{G}_n(\cdot,\cdot)$ on pavements $[0,k_n/n]^2$ of $[0,1]^2,$ where $k_n$ is a sequence of positive constants fulfilling some conditions. We provide a upper bound for the strong approximation of $\mathbb{G}_n(\cdot,\cdot)$ by a Gaussian process when $k_n/n \searrow \gamma$ as $n\rightarrow \infty,$ where $0 \leq \gamma \leq 1.$

We study the asymptotic of the spectral distribution for large empirical covariance matrices composed of independent Multifractal Random Walk processes. The asymptotic is taken as the observation lag shrinks to 0. In this setting, we show that there exists a limiting spectral distribution whose Stieltjes transform is uniquely characterized by equations which we specify. We also illustrate our results by numerical simulations.

The large deviations principle for the empirical measure for both continuous and discrete time Markov processes is well known. Various expressions are avail- able for the rate function, but these expressions are usually as the solution to a variational problem, and in this sense not explicit. An interesting class of con- tinuous time, reversible processes was identi ed in the original work of Donsker and Varadhan for which an explicit expression is possible. While this class in- cludes many (reversible) processes of interest, it excludes the case of continuous time pure jump processes, such as a reversible nite state Markov chain. In this paper we study the large deviations principle for the empirical measure of pure jump Markov processes and provide an explicit formula of the rate function under reversibility.

We derive strong approximations to the supremum of the non-centered empirical process indexed by a possibly unbounded VC-type class of functions by the suprema of the Gaussian and bootstrap processes. The bounds of these approximations are non-asymptotic, which allows us to work with classes of functions whose complexity increases with the sample size. The construction of couplings is not of the Hungarian type and is instead based on the Slepian-Stein methods and Gaussian comparison inequalities. The increasing complexity of classes of functions and non-centrality of the processes make the results useful for applications in modern nonparametric statistics (Gin\'{e} and Nickl, 2015), in particular allowing us to study the power properties of nonparametric tests using Gaussian and bootstrap approximations.

Journal of Research of the National Bureau of Standards

With the recent emergence of agile software development technologies like XP, many organizations, small and large, have adopted XP or a subset of its practices. This talk describes the empirical evaluation of the XP development process in terms of industrial case studies performed at IBM, Sabre and Tekelec. These case studies involve measurement of various product and process measures to compare the XP process and also discusses the obstacles that were encountered during the case studies. This talk also provides a brief comparative introduction between XP-1 and the recent XP-2 processes. ©2005 Microsoft Corporation. All rights reserved.

This research focused on the implementation of Open MP. It considers the parallelization of an application code that simulates the thermal gradient of material in two dimensions. A ‘C language’ program code called jacobi2d.c that solves a rectangular 2-dimensional heat conductivity problem using Jacobi iterative method was used. The boundary conditions required to compute a temperature distribution for a rectangular 2D problem are: Top 300C, Bottom 500C, Left 400C and Right 900C with a range of problem sizes enter as a run-time parameter to alter the problem sizes and convergence criteria. Also, there were computations and readings for iterations and runtime for four values of M and N which were selected for 01, 02, and 03 optimizations. In Table 1 Readings, four values were selected for each of the iterations. The results show the performance of the runtime as the processor increases from 01-optimization, to 02-optimization and finally to 03- optimization. It can be deduced from the representation that the run time of the values reduces as more resources are allocated to execution through the increase in optimization level. Also, in Table 2 Readings, the runtime decreases as it moves from thread1, thread2, thread3, and thread4, comparing the last values for thread1 which are M is 180, N is 200, and their runtime which is 42.797187001. Also the last values for thread2 which are M is 180, N is 200, their runtime which is 21.772106003. When the two runtimes were compared, it was discovered that there was a decrease in the runtime because the more the thread increases, the more system resources they share such as a processor which may affect their runtime by increasing it.

For a uniform process $\{ X_t: t\in E\}$ (by which $X_t $ is uniformly distributed on $(0,1)$ for $t\in E$) and a function $w(x)>0$ on $(0,1)$, we give a sufficient condition for the weak convergence of the empirical process based on $\{ w(x)(\mathbb{1}_{X_t\leq x} -x): t\in E, x\in [0,1]\}$ in $\ell^\infty(E\times [0,1])$. When specializing to $w(x)\equiv 1$ and assuming strict monotonicity on the marginal distribution functions of the input process, we recover a result of Kuelbs, Kurtz, and Zinn (2013). In the last section, we give an example of the main theorem.

By the use of a semi-martingale property of the Kolmogorov supremum, the results of Pyke on the weak convergence of the empirical process with random sample size are simplified and extended to the case of p(1)-dimensional stochastic vectors.

The empirical copula process plays a central role for statistical inference on copulas. Recently, Segers (2011) investigated the asymptotic behavior of this process under non-restrictive smoothness assumptions for the case of i.i.d. random variables. In the present paper we extend his main result to the case of serial dependent random variables by means of the powerful and elegant functional delta method. Moreover, we utilize the functional delta method in order to obtain conditional consistency of certain bootstrap procedures. Finally, we extend the results to the more general sequential empirical copula process under serial dependence.

In the present paper we obtain fully explicit large deviation inequalities for empirical processes indexed by a Vapnik--Chervonenkis class of sets (or functions). Furthermore we illustrate the importance of such results for the theory of information-based complexity.

While effective concentration inequalities for suprema of empirical processes exist under boundedness or strict tail assumptions, no comparable results have been available under considerably weaker assumptions. In this paper, we derive concentration inequalities assuming only low moments for an envelope of the empirical process. These concentration inequalities are beneficial even when the envelope is much larger than the single functions under consideration.

We study the empirical process indexed by F^2=\{f^2 : f \in F\}, where F is a class of mean-zero functions on a probability space. We present a sharp bound on the supremum of that process which depends on the \psi_1 diameter of the class F (rather than on the \psi_2 one) and on the complexity parameter \gamma_2(F,\psi_2). In addition, we present optimal bounds on the random diameters \sup_{f \in F} \max_{|I|=m} (\sum_{i \in I} f^2(X_i))^{1/2} using the same parameters. As applications, we extend several well known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on R^n.

We study the empirical process indexed by F^2=\{f^2 : f \in F\}, where F is a class of mean-zero functions on a probability space. We present a sharp bound on the supremum of that process which depends on the \psi_1 diameter of the class F (rather than on the \psi_2 one) and on the complexity parameter \gamma_2(F,\psi_2). In addition, we present optimal bounds on the random diameters \sup_{f \in F} \max_{|I|=m} (\sum_{i \in I} f^2(X_i))^{1/2} using the same parameters. As applications, we extend several well known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on R^n.

We study the empirical process indexed by F^2=\{f^2 : f \in F\}, where F is a class of mean-zero functions on a probability space. We present a sharp bound on the supremum of that process which depends on the \psi_1 diameter of the class F (rather than on the \psi_2 one) and on the complexity parameter \gamma_2(F,\psi_2). In addition, we present optimal bounds on the random diameters \sup_{f \in F} \max_{|I|=m} (\sum_{i \in I} f^2(X_i))^{1/2} using the same parameters. As applications, we extend several well known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on R^n.

In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical "unbiased behavior" with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper we prove, using ergodic theory and the notion of category, that clustering (even very strong) is in fact typical for "rare events" defined as long cylinder sets in processes generated by a finite partition of an arbitrary (infinite aperiodic) ergodic measure preserving transformation.

For fixed $t\in [0,1)$ and $h>0$, consider the local uniform empirical process $$\DD_{n,h,t}(s):=n^{-1/2}\coo\sliin 1_{[t,t+hs]}(U_i)-hs\cff,\;s\in [0,1],$$ where the $U_i$ are independent and uniformly distributed on $[0,1]$. We investigate the functional limit behaviour of $\DD_{n,h,t}$ uniformly in $\wth_n\le h\le h_n$ when $n\wth_n/\log\log n\rar \infty$ and $h_n\rar 0$.

We investigate the convergence in distribution of sequential empirical processes of dependent data indexed by a class of functions F. Our technique is suitable for processes that satisfy a multiple mixing condition on a space of functions which differs from the class F. This situation occurs in the case of data arising from dynamical systems or Markov chains, for which the Perron--Frobenius or Markov operator, respectively, has a spectral gap on a restricted space. We provide applications to iterative Lipschitz models that contract on average.

We investigate the convergence in distribution of sequential empirical processes of dependent data indexed by a class of functions F. Our technique is suitable for processes that satisfy a multiple mixing condition on a space of functions which differs from the class F. This situation occurs in the case of data arising from dynamical systems or Markov chains, for which the Perron--Frobenius or Markov operator, respectively, has a spectral gap on a restricted space. We provide applications to iterative Lipschitz models that contract on average.

A unified power-law relationship between average erosion rate and cumulative erosion is presented. Extensive data analyses from venturi, magnetostriction (stationary and oscillating specimens), liquid drop, and jet impact devices appear to conform to this relation. A normalization technique using cavitation and liquid impingement erosion data is also presented to facilitate prediction. Attempts are made to understand the relationship between the coefficients in the power-law relationships and the material properties.

A moderate deviation principle for functionals, with at most quadratic growth, of moving average processes is established. The main assumptions on the moving average process are a Logarithmic Sobolev inequality for the driving random variables and the continuity, or weaker, of the spectral density of the moving average process. We also obtain the moderate deviations for the empirical spectral density, exhibiting an interesting new form of the rate function, i.e. with a correction term compared to the Gaussian rate functionnal.

We study the asymptotic behavior of empirical processes generated by measurable bounded functions of an infinite source Poisson transmission process when the session length have infinite variance. In spite of the boundedness of the function, the normalized fluctuations of such an empirical process converge to a non-Gaussian stable process. This phenomenon can be viewed as caused by the long-range dependence in the transmission process. Completing previous results on the empirical mean of similar types of processes, our results on non-linear bounded functions exhibit the influence of the limit transmission rate distribution at high session lengths on the asymptotic behavior of the empirical process. As an illustration, we apply the main result to estimation of the distribution function of the steady state value of the transmission process.

We introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional. As an application we give a new easy proof of Talagrand's concentration inequality for empirical processes, where besides symmetrization we use only Talagrand's concentration inequality on the discrete cube {-1,+1}^n. As another application of this technique we prove new Vapnik-Chervonenkis type inequalities. For example, for VC-classes of functions we prove a classical inequality of Vapnik and Chervonenkis only with normalization by the sum of variance and sample variance.

This scoping review is intended to systematically examine the evidence for and against the assumption of an ability to access and explain one’s own memory, motivations, and decision making.

This scoping review is intended to systematically examine the evidence for and against the assumption of an ability to access and explain one’s own memory, motivations, and decision making.

Following the works of Berthet (1997), we first obtain exact clustering rates in the functional law of the iterated logarithm for the uniform empirical and quantile processes and for their increments. In a second time, we obtain functional Chung-type limit laws for the local empirical process for a class of target functions on the border of the Strassen set.

Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube. The assumption is non-restrictive in the sense that it is needed anyway to ensure that the candidate limiting process exists and has continuous trajectories. In addition, resampling methods based on the multiplier central limit theorem, which require consistent estimation of the first-order derivatives, continue to be valid. Under certain growth conditions on the second-order partial derivatives that allow for explosive behavior near the boundaries, the almost sure rate in Stute's representation of the empirical copula process can be recovered. The conditions are verified, for instance, in the case of the Gaussian copula with full-rank correlation matrix, many Archimedean copulas, and many extreme-value copulas.

We give several conditions for pregaussianity of norm balls of Besov spaces defined over $\mathbb{R}^d$ by exploiting results in Haroske and Triebel (2005). Furthermore, complementing sufficient conditions in Nickl and P\"{o}tscher (2005), we give necessary conditions on the parameters of the Besov space to obtain the Donsker property of such balls. For certain parameter combinations Besov balls are shown to be pregaussian but not Donsker.

We consider marked empirical processes, indexed by a randomly projected functional covariate, to construct goodness-of-fit tests for the functional linear model with scalar response. The test statistics are built from continuous functionals over the projected process, resulting in computationally efficient tests that exhibit root-n convergence rate and circumvent the curse of dimensionality. The weak convergence of the process is obtained conditionally on a random direction, whilst it is proved the almost surely equivalence between the testing for significance expressed on the original and on the projected functional covariate. The computation of the test in practise involves the calibration by wild bootstrap resampling and the combination of several p-values arising from different projections by means of the false discovery rate method. The finite sample properties of the test are illustrated in a simulation study for a variety of linear models, underlying processes and alternatives. The software provided implements the tests and allows the replication of simulations and data applications.

Research planned to improve understanding of conative educational processes and outcomes is described. Over the past year, literature reviews have been conducted aimed at defining categories of assessment of student persistence, freedom from distraction, engagement, and other school-and-work related attitudes and motivations, both as learning processes and instructional outcomes. These conative processes and outcomes are considered distinct from cognitive measures and characteristics. Study 1 will examine the attitudes of about 200 high school students using a conative questionnaire. In Study 2, about 30 students will be selected from the first study to complete additional tasks (a simulated video game and computer teaching task) and an interview. Assessments will consider action versus state orientation, mastery versus performance orientation, and mindfulness. A table lists estimated time requirements for the studies. (SLD)

We correct an error in a technical lemma of Drees and Rootz\'en (2010) [arXiv:0910.0343] and discuss consequences for applications.

The primary goal of distributed discrete event simulations is to achieve speedup in simulation execution time by distributing the processing of the simulation over multiple processors. When partitioned for distribution in this fashion, simulations are typically partitioned such that there are more processes than processors. This thesis reviews existing methods for distributed discrete event simulations, and proposes general guidelines for efficient partitioning for a given communications topology based on empirical evidence. A performance analysis is conducted for two approaches to partitioning the system. The first method chosen is a mapping of multiple processes to a processor and the second approach utilizes a distributed event list approach, developed by Mannix. This approach combines smaller processes into a larger single process, incorporating a next event list similar to that used in a sequential simulation. Empirical studies compare the performance of the two approaches under a variety of conditions. The traditional Chandy-Mirsa approach to system partitioning is demonstrated to yield overall better performance than the distributed event list algorithm. General guidelines for partitioning the system for both approaches are developed based on the performance comparisons.

The large deviations principle for the empirical measure for both continuous and discrete time Markov processes is well known. Various expressions are available for the rate function, but these expressions are usually as the solution to a variational problem, and in this sense not explicit. An interesting class of continuous time, reversible processes was identified in the original work of Donsker and Varadhan for which an explicit expression is possible. While this class includes many (reversible) processes of interest, it excludes the case of continuous time pure jump processes, such as a reversible finite state Markov chain. In this paper we study the large deviations principle for the empirical measure of pure jump Markov processes and provide an explicit formula of the rate function under reversibility.

Empirical processes for stationary, causal sequences are considered. We establish empirical central limit theorems for classes of indicators of left half lines, absolutely continuous functions and piecewise differentiable functions. Sample path properties of empirical distribution functions are also discussed. The results are applied to linear processes and Markov chains.

Empirical processes for stationary, causal sequences are considered. We establish empirical central limit theorems for classes of indicators of left half lines, absolutely continuous functions and piecewise differentiable functions. Sample path properties of empirical distribution functions are also discussed. The results are applied to linear processes and Markov chains.

This paper studies the residual empirical process of long- and short-memory time series regression models and establishes its uniform expansion under a general framework. The results are applied to the stochastic regression models and unstable autoregressive models. For the long-memory noise, it is shown that the limit distribution of the Kolmogorov-Smirnov test statistic studied in Ho and Hsing [Ann. Statist. 24 (1996) 992-1024] does not hold when the stochastic regression model includes an unknown intercept or when the characteristic polynomial of the unstable autoregressive model has a unit root. To this end, two new statistics are proposed to test for the distribution of the long-memory noises of stochastic regression models and unstable autoregressive models. (With Correction.)