Point Processes
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Poisson Processes Stochastic Processes
Poisson Processes Stochastic Processes UC3M Feb. 2012 Exponential random variables A random variable T has exponential distribution with rate λ > 0 if its probability density function can been written as −λt f (t) = λe 1(0;+1)(t) We summarize the above by T ∼ exp(λ): The cumulative distribution function of a exponential random variable is −λt F (t) = P(T ≤ t) = 1 − e 1(0;+1)(t) And the tail, expectation and variance are P(T > t) = e−λt ; E[T ] = λ−1; and Var(T ) = E[T ] = λ−2 The exponential random variable has the lack of memory property P(T > t + sjT > t) = P(T > s) Exponencial races In what follows, T1;:::; Tn are independent r.v., with Ti ∼ exp(λi ). P1: min(T1;:::; Tn) ∼ exp(λ1 + ··· + λn) . P2 λ1 P(T1 < T2) = λ1 + λ2 P3: λi P(Ti = min(T1;:::; Tn)) = λ1 + ··· + λn P4: If λi = λ and Sn = T1 + ··· + Tn ∼ Γ(n; λ). That is, Sn has probability density function (λs)n−1 f (s) = λe−λs 1 (s) Sn (n − 1)! (0;+1) The Poisson Process as a renewal process Let T1; T2;::: be a sequence of i.i.d. nonnegative r.v. (interarrival times). Define the arrival times Sn = T1 + ··· + Tn if n ≥ 1 and S0 = 0: The process N(t) = maxfn : Sn ≤ tg; is called Renewal Process. If the common distribution of the times is the exponential distribution with rate λ then process is called Poisson Process of with rate λ. Lemma. N(t) ∼ Poisson(λt) and N(t + s) − N(s); t ≥ 0; is a Poisson process independent of N(s); t ≥ 0 The Poisson Process as a L´evy Process A stochastic process fX (t); t ≥ 0g is a L´evyProcess if it verifies the following properties: 1. -
The Maximum of a Random Walk Reflected at a General Barrier
The Annals of Applied Probability 2006, Vol. 16, No. 1, 15–29 DOI: 10.1214/105051605000000610 c Institute of Mathematical Statistics, 2006 THE MAXIMUM OF A RANDOM WALK REFLECTED AT A GENERAL BARRIER By Niels Richard Hansen University of Copenhagen We define the reflection of a random walk at a general barrier and derive, in case the increments are light tailed and have negative mean, a necessary and sufficient criterion for the global maximum of the reflected process to be finite a.s. If it is finite a.s., we show that the tail of the distribution of the global maximum decays exponentially fast and derive the precise rate of decay. Finally, we discuss an example from structural biology that motivated the interest in the reflection at a general barrier. 1. Introduction. The reflection of a random walk at zero is a well-studied process with several applications. We mention the interpretation from queue- ing theory—for a suitably defined random walk—as the waiting time until service for a customer at the time of arrival; see, for example, [1]. Another important application arises in molecular biology in the context of local com- parison of two finite sequences. To evaluate the significance of the findings from such a comparison, one needs to study the distribution of the locally highest scoring segment from two independent i.i.d. sequences, as shown in [8], which equals the distribution of the maximum of a random walk reflected at zero. The global maximum of a random walk with negative drift and, in par- ticular, the probability that the maximum exceeds a high value have also been studied in details. -
Renewal Theory for Uniform Random Variables
California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2002 Renewal theory for uniform random variables Steven Robert Spencer Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Spencer, Steven Robert, "Renewal theory for uniform random variables" (2002). Theses Digitization Project. 2248. https://scholarworks.lib.csusb.edu/etd-project/2248 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. RENEWAL THEORY FOR UNIFORM RANDOM VARIABLES A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Steven Robert Spencer March 2002 RENEWAL THEORY FOR UNIFORM RANDOM VARIABLES A Thesis Presented to the Faculty of California State University, San Bernardino by Steven Robert Spencer March 2002 Approved by: Charles Stanton, Advisor, Mathematics Date Yuichiro Kakihara Terry Hallett j. Peter Williams, Chair Terry Hallett, Department of Mathematics Graduate Coordinator Department of Mathematics ABSTRACT The thesis answers the question, "How many times must you change a light-bulb in a month if the life-time- of any one light-bulb is anywhere from zero to one month in length?" This involves uniform random variables on the - interval [0,1] which must be summed to give expected values for the problem. The results of convolution calculations for both the uniform and exponential distributions of random variables give expected values that are in accordance with the Elementary Renewal Theorem and renewal function. -
Completely Random Measures and Related Models
CRMs Sinead Williamson Background Completely random measures and related L´evyprocesses Completely models random measures Applications Normalized Sinead Williamson random measures Neutral-to-the- right processes Computational and Biological Learning Laboratory Exchangeable University of Cambridge matrices January 20, 2011 Outline CRMs Sinead Williamson 1 Background Background L´evyprocesses Completely 2 L´evyprocesses random measures Applications 3 Completely random measures Normalized random measures Neutral-to-the- right processes 4 Applications Exchangeable matrices Normalized random measures Neutral-to-the-right processes Exchangeable matrices A little measure theory CRMs Sinead Williamson Set: e.g. Integers, real numbers, people called James. Background May be finite, countably infinite, or uncountably infinite. L´evyprocesses Completely Algebra: Class T of subsets of a set T s.t. random measures 1 T 2 T . 2 If A 2 T , then Ac 2 T . Applications K Normalized 3 If A1;:::; AK 2 T , then [ Ak = A1 [ A2 [ ::: AK 2 T random k=1 measures (closed under finite unions). Neutral-to-the- right K processes 4 If A1;:::; AK 2 T , then \k=1Ak = A1 \ A2 \ ::: AK 2 T Exchangeable matrices (closed under finite intersections). σ-Algebra: Algebra that is closed under countably infinite unions and intersections. A little measure theory CRMs Sinead Williamson Background L´evyprocesses Measurable space: Combination (T ; T ) of a set and a Completely σ-algebra on that set. random measures Measure: Function µ between a σ-field and the positive Applications reals (+ 1) s.t. Normalized random measures 1 µ(;) = 0. Neutral-to-the- right 2 For all countable collections of disjoint sets processes P Exchangeable A1; A2; · · · 2 T , µ([k Ak ) = µ(Ak ). -
POISSON PROCESSES 1.1. the Rutherford-Chadwick-Ellis
POISSON PROCESSES 1. THE LAW OF SMALL NUMBERS 1.1. The Rutherford-Chadwick-Ellis Experiment. About 90 years ago Ernest Rutherford and his collaborators at the Cavendish Laboratory in Cambridge conducted a series of pathbreaking experiments on radioactive decay. In one of these, a radioactive substance was observed in N = 2608 time intervals of 7.5 seconds each, and the number of decay particles reaching a counter during each period was recorded. The table below shows the number Nk of these time periods in which exactly k decays were observed for k = 0,1,2,...,9. Also shown is N pk where k pk = (3.87) exp 3.87 =k! {− g The parameter value 3.87 was chosen because it is the mean number of decays/period for Rutherford’s data. k Nk N pk k Nk N pk 0 57 54.4 6 273 253.8 1 203 210.5 7 139 140.3 2 383 407.4 8 45 67.9 3 525 525.5 9 27 29.2 4 532 508.4 10 16 17.1 5 408 393.5 ≥ This is typical of what happens in many situations where counts of occurences of some sort are recorded: the Poisson distribution often provides an accurate – sometimes remarkably ac- curate – fit. Why? 1.2. Poisson Approximation to the Binomial Distribution. The ubiquity of the Poisson distri- bution in nature stems in large part from its connection to the Binomial and Hypergeometric distributions. The Binomial-(N ,p) distribution is the distribution of the number of successes in N independent Bernoulli trials, each with success probability p. -
Central Limit Theorem for Supercritical Binary Homogeneous Crump-Mode
Central limit theorem for supercritical binary homogeneous Crump-Mode-Jagers processes Benoit Henry1,2 Abstract We consider a supercritical general branching population where the lifetimes of individuals are i.i.d. with arbitrary distribution and each individual gives birth to new individuals at Poisson times independently from each others. The population counting process of such population is a known as binary homogeneous Crump-Jargers-Mode process. It is known that such processes converges almost surely when correctly renormalized. In this paper, we study the error of this convergence. To this end, we use classical renewal theory and recent works [17, 6, 5] on this model to obtain the moments of the error. Then, we can precisely study the asymptotic behaviour of these moments thanks to L´evy processes theory. These results in conjunction with a new decomposition of the splitting trees allow us to obtain a central limit theorem. MSC 2000 subject classifications: Primary 60J80; secondary 92D10, 60J85, 60G51, 60K15, 60F05. Key words and phrases. branching process – splitting tree – Crump–Mode–Jagers process – linear birth–death process – L´evy processes – scale function – Central Limit Theorem. 1 Introduction In this work, we consider a general branching population where individuals live and reproduce inde- pendently from each other. Their lifetimes follow an arbitrary distribution PV and the births occur at Poisson times with constant rate b. The genealogical tree induced by this population is called a arXiv:1509.06583v2 [math.PR] 18 Nov 2016 splitting tree [11, 10, 17] and is of main importance in the study of the model. The population counting process Nt (giving the number of living individuals at time t) is a binary homogeneous Crump-Mode-Jagers (CMJ) process. -
Some Theory for the Analysis of Random Fields Diplomarbeit
Philipp Pluch Some Theory for the Analysis of Random Fields With Applications to Geostatistics Diplomarbeit zur Erlangung des akademischen Grades Diplom-Ingenieur Studium der Technischen Mathematik Universit¨at Klagenfurt Fakult¨at fur¨ Wirtschaftswissenschaften und Informatik Begutachter: O.Univ.-Prof.Dr. Jurgen¨ Pilz Institut fur¨ Mathematik September 2004 To my parents, Verena and all my friends Ehrenw¨ortliche Erkl¨arung Ich erkl¨are ehrenw¨ortlich, dass ich die vorliegende Schrift verfasst und die mit ihr unmittelbar verbundenen Arbeiten selbst durchgefuhrt¨ habe. Die in der Schrift verwendete Literatur sowie das Ausmaß der mir im gesamten Arbeitsvorgang gew¨ahrten Unterstutzung¨ sind ausnahmslos angegeben. Die Schrift ist noch keiner anderen Prufungsb¨ eh¨orde vorgelegt worden. St. Urban, 29 September 2004 Preface I remember when I first was at our univeristy - I walked inside this large corridor called ’Aula’ and had no idea what I should do, didn’t know what I should study, I had interest in Psychology or Media Studies, and now I’m sitting in my office at the university, five years later, writing my final lines for my master degree theses in mathematics. A long and also hard but so beautiful way was gone, I remember at the beginning, the first mathematic courses in discrete mathematics, how difficult that was for me, the abstract thinking, my first exams and now I have finished them all, I mastered them. I have to thank so many people and I will do so now. First I have to thank my parents, who always believed in me, who gave me financial support and who had to fight with my mood when I was working hard. -
Completely Random Measures
Pacific Journal of Mathematics COMPLETELY RANDOM MEASURES JOHN FRANK CHARLES KINGMAN Vol. 21, No. 1 November 1967 PACIFIC JOURNAL OF MATHEMATICS Vol. 21, No. 1, 1967 COMPLETELY RANDOM MEASURES J. F. C. KlNGMAN The theory of stochastic processes is concerned with random functions defined on some parameter set. This paper is con- cerned with the case, which occurs naturally in some practical situations, in which the parameter set is a ^-algebra of subsets of some space, and the random functions are all measures on this space. Among all such random measures are distinguished some which are called completely random, which have the property that the values they take on disjoint subsets are independent. A representation theorem is proved for all completely random measures satisfying a weak finiteness condi- tion, and as a consequence it is shown that all such measures are necessarily purely atomic. 1. Stochastic processes X(t) whose realisation are nondecreasing functions of a real parameter t occur in a number of applications of probability theory. For instance, the number of events of a point process in the interval (0, t), the 'load function' of a queue input [2], the amount of water entering a reservoir in time t, and the local time process in a Markov chain ([8], [7] §14), are all processes of this type. In many applications the function X(t) enters as a convenient way of representing a measure on the real line, the Stieltjes measure Φ defined as the unique Borel measure for which (1) Φ(a, b] = X(b + ) - X(a + ), (-co <α< b< oo). -
Applications of Renewal Theory to Pattern Analysis
Rochester Institute of Technology RIT Scholar Works Theses 4-18-2016 Applications of Renewal Theory to Pattern Analysis Hallie L. Kleiner [email protected] Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Kleiner, Hallie L., "Applications of Renewal Theory to Pattern Analysis" (2016). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. APPLICATIONS OF RENEWAL THEORY TO PATTERN ANALYSIS by Hallie L. Kleiner A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Applied Mathematics School of Mathematical Sciences, College of Science Rochester Institute of Technology Rochester, NY April 18, 2016 Committee Approval: Dr. James Marengo Date School of Mathematical Sciences Thesis Advisor Dr. Bernard Brooks Date School of Mathematical Sciences Committee Member Dr. Manuel Lopez Date School of Mathematical Sciences Committee Member Dr. Elizabeth Cherry Date School of Mathematical Sciences Director of Graduate Programs RENEWAL THEORY ON PATTERN ANALYSIS Abstract In this thesis, renewal theory is used to analyze patterns of outcomes for discrete random variables. We start by introducing the concept of a renewal process and by giving some examples. It is then shown that we may obtain the distribution of the number of renewals by time t and compute its expectation, the renewal function. We then proceed to state and illustrate the basic limit theorems for renewal processes and make use of Wald’s equation to give a proof of the Elementary Renewal Theorem. -
Rescaling Marked Point Processes
1 Rescaling Marked Point Processes David Vere-Jones1, Victoria University of Wellington Frederic Paik Schoenberg2, University of California, Los Angeles Abstract In 1971, Meyer showed how one could use the compensator to rescale a multivari- ate point process, forming independent Poisson processes with intensity one. Meyer’s result has been generalized to multi-dimensional point processes. Here, we explore gen- eralization of Meyer’s theorem to the case of marked point processes, where the mark space may be quite general. Assuming simplicity and the existence of a conditional intensity, we show that a marked point process can be transformed into a compound Poisson process with unit total rate and a fixed mark distribution. ** Key words: residual analysis, random time change, Meyer’s theorem, model evaluation, intensity, compensator, Poisson process. ** 1 Department of Mathematics, Victoria University of Wellington, P.O. Box 196, Welling- ton, New Zealand. [email protected] 2 Department of Statistics, 8142 Math-Science Building, University of California, Los Angeles, 90095-1554. [email protected] Vere-Jones and Schoenberg. Rescaling Marked Point Processes 2 1 Introduction. Before other matters, both authors would like express their appreciation to Daryl for his stimulating and forgiving company, and to wish him a long and fruitful continuation of his mathematical, musical, woodworking, and many other activities. In particular, it is a real pleasure for the first author to acknowledge his gratitude to Daryl for his hard work, good humour, generosity and continuing friendship throughout the development of (innumerable draft versions and now even two editions of) their joint text on point processes. -
Lecture 26 : Poisson Point Processes
Lecture 26 : Poisson Point Processes STAT205 Lecturer: Jim Pitman Scribe: Ben Hough <[email protected]> In this lecture, we consider a measure space (S; S; µ), where µ is a σ-finite measure (i.e. S may be written as a disjoint union of sets of finite µ-measure). 26.1 The Poisson Point Process Definition 26.1 A Poisson Point Process (P.P.P.) with intensity µ is a collection of random variables N(A; !), A 2 S, ! 2 Ω defined on a probability space (Ω; F; P) such that: 1. N(·; !) is a counting measure on (S; S) for each ! 2 Ω. 2. N(A; ·) is Poisson with mean µ(A): − P e µ(A)(µ(A))k (N(A) = k) = k! all A 2 S. 3. If A1, A2, ... are disjoint sets then N(A1; ·), N(A2; ·), ... are independent random variables. Theorem 26.2 P.P.P.'s exist. Proof Sketch: It's not enough to quote Kolmogorov's Extension Theorem. The only convincing argument is to give an explicit constuction from sequences of independent random variables. We begin by considering the case µ(S) < 1. P µ(A) 1. Take X1, X2, ... to be i.i.d. random variables so that (Xi 2 A) = µ(S) . 2. Take N(S) to be a Poisson random variable with mean µ(S), independent of the Xi's. Assume all random varibles are defined on the same probability space (Ω; F; P). N(S) 3. Define N(A) = i=1 1(Xi2A), for all A 2 S. P 1 Now verify that this N(A; !) is a P.P.P. -
Renewal Monte Carlo: Renewal Theory Based Reinforcement Learning Jayakumar Subramanian and Aditya Mahajan
1 Renewal Monte Carlo: Renewal theory based reinforcement learning Jayakumar Subramanian and Aditya Mahajan Abstract—In this paper, we present an online reinforcement discounted and average reward setups as well as for models learning algorithm, called Renewal Monte Carlo (RMC), for with continuous state and action spaces. However, they suffer infinite horizon Markov decision processes with a designated from various drawbacks. First, they have a high variance start state. RMC is a Monte Carlo algorithm and retains the advantages of Monte Carlo methods including low bias, simplicity, because a single sample path is used to estimate performance. and ease of implementation while, at the same time, circumvents Second, they are not asymptotically optimal for infinite horizon their key drawbacks of high variance and delayed (end of models because it is effectively assumed that the model is episode) updates. The key ideas behind RMC are as follows. episodic; in infinite horizon models, the trajectory is arbitrarily First, under any reasonable policy, the reward process is ergodic. truncated to treat the model as an episodic model. Third, the So, by renewal theory, the performance of a policy is equal to the ratio of expected discounted reward to the expected policy improvement step cannot be carried out in tandem discounted time over a regenerative cycle. Second, by carefully with policy evaluation. One must wait until the end of the examining the expression for performance gradient, we propose a episode to estimate the performance and only then can the stochastic approximation algorithm that only requires estimates policy parameters be updated. It is for these reasons that of the expected discounted reward and discounted time over a Monte Carlo methods are largely ignored in the literature on regenerative cycle and their gradients.