On a class of reflected AR(1) processes
TL;DR: In this article, it was shown that the reflected AR(1) process converges to a reflected Ornstein-Uhlenbeck process under heavy traffic scaling, and the corresponding steady-state distribution converged to the distribution of a normal random variable conditioned on being positive.
Abstract: In this paper we study a reflected AR(1) process, i.e. a process (Z(n))(n) obeying the recursion Z(n+1) = max{aZ(n) + X-n, 0}, with (X-n)(n) a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Z(n) (in terms of transforms) in case X-n can be written as Y-n - B-n, with (B-n) n being a sequence of independent random variables which are all Exp(lambda) distributed, and (Y-n)(n) i.i.d.; when vertical bar a vertical bar < 1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (B-n)(n) are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein-Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.
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01 Jan 2011
TL;DR: Weakconvergence methods in metric spaces were studied in this article, with applications sufficient to show their power and utility, and the results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables.
Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an
3,554 citations
TL;DR: In this paper, applied probability and queuing in the field of applied probabilistic analysis is discussed. But the authors focus on the application of queueing in the context of road traffic.
Abstract: (1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.
1,121 citations
[...]
TL;DR: In this paper, the authors pointed out that the distinction between "finite" and "infinite" is one which does not require definition, and that the authors' view is not the only accepted view.
Abstract: THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.
740 citations
TL;DR: A general class of storage processes, in which the rate at which the storage level increases or decreases is assumed to be an affine function of the current storage level, and, in addition, both upward and downward jumps are allowed, is analyzed.
Abstract: The aim of this paper is to analyze a general class of storage processes, in which the rate at which the storage level increases or decreases is assumed to be an affine function of the current stor
16 citations
TL;DR: The two processes INGAR+ and GAR+ are shown to be connected via a duality relation and a detailed analysis of the time-dependent and stationary behavior of the INGar+ process is presented.
Abstract: We introduce two general classes of reflected autoregressive processes, INGAR+ and GAR+. Here, INGAR+ can be seen as the counterpart of INAR(1) with general thinning and reflection being imposed to keep the process non-negative; GAR+ relates to AR(1) in an analogous manner. The two processes INGAR+ and GAR+ are shown to be connected via a duality relation. We proceed by presenting a detailed analysis of the time-dependent and stationary behavior of the INGAR+ process, and then exploit the duality relation to obtain the time-dependent and stationary behavior of the GAR+ process.
12 citations
References
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Book•
01 Jan 1968
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Abstract: Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.
13,153 citations
TL;DR: A general approach to Time Series Modelling and ModeLLing with ARMA Processes, which describes the development of a Stationary Process in Terms of Infinitely Many Past Values and the Autocorrelation Function.
Abstract: Preface 1 INTRODUCTION 1.1 Examples of Time Series 1.2 Objectives of Time Series Analysis 1.3 Some Simple Time Series Models 1.3.3 A General Approach to Time Series Modelling 1.4 Stationary Models and the Autocorrelation Function 1.4.1 The Sample Autocorrelation Function 1.4.2 A Model for the Lake Huron Data 1.5 Estimation and Elimination of Trend and Seasonal Components 1.5.1 Estimation and Elimination of Trend in the Absence of Seasonality 1.5.2 Estimation and Elimination of Both Trend and Seasonality 1.6 Testing the Estimated Noise Sequence 1.7 Problems 2 STATIONARY PROCESSES 2.1 Basic Properties 2.2 Linear Processes 2.3 Introduction to ARMA Processes 2.4 Properties of the Sample Mean and Autocorrelation Function 2.4.2 Estimation of $\gamma(\cdot)$ and $\rho(\cdot)$ 2.5 Forecasting Stationary Time Series 2.5.3 Prediction of a Stationary Process in Terms of Infinitely Many Past Values 2.6 The Wold Decomposition 1.7 Problems 3 ARMA MODELS 3.1 ARMA($p,q$) Processes 3.2 The ACF and PACF of an ARMA$(p,q)$ Process 3.2.1 Calculation of the ACVF 3.2.2 The Autocorrelation Function 3.2.3 The Partial Autocorrelation Function 3.3 Forecasting ARMA Processes 1.7 Problems 4 SPECTRAL ANALYSIS 4.1 Spectral Densities 4.2 The Periodogram 4.3 Time-Invariant Linear Filters 4.4 The Spectral Density of an ARMA Process 1.7 Problems 5 MODELLING AND PREDICTION WITH ARMA PROCESSES 5.1 Preliminary Estimation 5.1.1 Yule-Walker Estimation 5.1.3 The Innovations Algorithm 5.1.4 The Hannan-Rissanen Algorithm 5.2 Maximum Likelihood Estimation 5.3 Diagnostic Checking 5.3.1 The Graph of $\t=1,\ldots,n\ 5.3.2 The Sample ACF of the Residuals
3,732 citations
Book•
01 Jan 1987
TL;DR: In this paper, a simple Markovian model for queueing theory at the Markovians level is proposed, which is based on the theory of random walks and single server queueing.
Abstract: Preface SIMPLE MARKOVIAN MODELS: Markov Chains Markov Jump Processes Queueing Theory at the Markovian Level BASIC MATHEMATICAL TOOLS: Basic Renewal Theory Regenerative Processes Further Topics in Renewal Theory and Regenerative Processes Random Walks SPECIAL MODELS AND METHODS: Steady-state Properties of GI/G/1 Explicit Examples in the Theory of Random Walks and Single Server Queues Multi-Dimensional Methods Many-server Queues Conjugate Processes Insurance Risk, Dam and Storage Models Selected Background and Notation.
2,757 citations
Book•
01 Jan 1932
TL;DR: Alfaro et al. as mentioned in this paper conservado en la Biblioteca del Campus de Mostoles de la Universidad Rey Juan Carlos (sign. 517.5 TIT THE).
Abstract: Original conservado en la Biblioteca del Campus de Mostoles de la Universidad Rey Juan Carlos (sign. 517.5 TIT THE).
2,695 citations
Book•
01 Jan 1991TL;DR: A branching-process example and an easy strong law: product measure using martingale theory and the central limit theorem are presented.
Abstract: Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital role. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.
2,265 citations