Stochastic Geometry and Wireless Networks: Volume I Theory Contents
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Poisson Representations of Branching Markov and Measure-Valued
The Annals of Probability 2011, Vol. 39, No. 3, 939–984 DOI: 10.1214/10-AOP574 c Institute of Mathematical Statistics, 2011 POISSON REPRESENTATIONS OF BRANCHING MARKOV AND MEASURE-VALUED BRANCHING PROCESSES By Thomas G. Kurtz1 and Eliane R. Rodrigues2 University of Wisconsin, Madison and UNAM Representations of branching Markov processes and their measure- valued limits in terms of countable systems of particles are con- structed for models with spatially varying birth and death rates. Each particle has a location and a “level,” but unlike earlier con- structions, the levels change with time. In fact, death of a particle occurs only when the level of the particle crosses a specified level r, or for the limiting models, hits infinity. For branching Markov pro- cesses, at each time t, conditioned on the state of the process, the levels are independent and uniformly distributed on [0,r]. For the limiting measure-valued process, at each time t, the joint distribu- tion of locations and levels is conditionally Poisson distributed with mean measure K(t) × Λ, where Λ denotes Lebesgue measure, and K is the desired measure-valued process. The representation simplifies or gives alternative proofs for a vari- ety of calculations and results including conditioning on extinction or nonextinction, Harris’s convergence theorem for supercritical branch- ing processes, and diffusion approximations for processes in random environments. 1. Introduction. Measure-valued processes arise naturally as infinite sys- tem limits of empirical measures of finite particle systems. A number of ap- proaches have been developed which preserve distinct particles in the limit and which give a representation of the measure-valued process as a transfor- mation of the limiting infinite particle system. -
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. -
A Stochastic Processes and Martingales
A Stochastic Processes and Martingales A.1 Stochastic Processes Let I be either IINorIR+.Astochastic process on I with state space E is a family of E-valued random variables X = {Xt : t ∈ I}. We only consider examples where E is a Polish space. Suppose for the moment that I =IR+. A stochastic process is called cadlag if its paths t → Xt are right-continuous (a.s.) and its left limits exist at all points. In this book we assume that every stochastic process is cadlag. We say a process is continuous if its paths are continuous. The above conditions are meant to hold with probability 1 and not to hold pathwise. A.2 Filtration and Stopping Times The information available at time t is expressed by a σ-subalgebra Ft ⊂F.An {F ∈ } increasing family of σ-algebras t : t I is called a filtration.IfI =IR+, F F F we call a filtration right-continuous if t+ := s>t s = t. If not stated otherwise, we assume that all filtrations in this book are right-continuous. In many books it is also assumed that the filtration is complete, i.e., F0 contains all IIP-null sets. We do not assume this here because we want to be able to change the measure in Chapter 4. Because the changed measure and IIP will be singular, it would not be possible to extend the new measure to the whole σ-algebra F. A stochastic process X is called Ft-adapted if Xt is Ft-measurable for all t. If it is clear which filtration is used, we just call the process adapted.The {F X } natural filtration t is the smallest right-continuous filtration such that X is adapted. -
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. -
Introduction to Stochastic Processes - Lecture Notes (With 33 Illustrations)
Introduction to Stochastic Processes - Lecture Notes (with 33 illustrations) Gordan Žitković Department of Mathematics The University of Texas at Austin Contents 1 Probability review 4 1.1 Random variables . 4 1.2 Countable sets . 5 1.3 Discrete random variables . 5 1.4 Expectation . 7 1.5 Events and probability . 8 1.6 Dependence and independence . 9 1.7 Conditional probability . 10 1.8 Examples . 12 2 Mathematica in 15 min 15 2.1 Basic Syntax . 15 2.2 Numerical Approximation . 16 2.3 Expression Manipulation . 16 2.4 Lists and Functions . 17 2.5 Linear Algebra . 19 2.6 Predefined Constants . 20 2.7 Calculus . 20 2.8 Solving Equations . 22 2.9 Graphics . 22 2.10 Probability Distributions and Simulation . 23 2.11 Help Commands . 24 2.12 Common Mistakes . 25 3 Stochastic Processes 26 3.1 The canonical probability space . 27 3.2 Constructing the Random Walk . 28 3.3 Simulation . 29 3.3.1 Random number generation . 29 3.3.2 Simulation of Random Variables . 30 3.4 Monte Carlo Integration . 33 4 The Simple Random Walk 35 4.1 Construction . 35 4.2 The maximum . 36 1 CONTENTS 5 Generating functions 40 5.1 Definition and first properties . 40 5.2 Convolution and moments . 42 5.3 Random sums and Wald’s identity . 44 6 Random walks - advanced methods 48 6.1 Stopping times . 48 6.2 Wald’s identity II . 50 6.3 The distribution of the first hitting time T1 .......................... 52 6.3.1 A recursive formula . 52 6.3.2 Generating-function approach . -
Introduction to Lévy Processes
Introduction to L´evyprocesses Graduate lecture 22 January 2004 Matthias Winkel Departmental lecturer (Institute of Actuaries and Aon lecturer in Statistics) 1. Random walks and continuous-time limits 2. Examples 3. Classification and construction of L´evy processes 4. Examples 5. Poisson point processes and simulation 1 1. Random walks and continuous-time limits 4 Definition 1 Let Yk, k ≥ 1, be i.i.d. Then n X 0 Sn = Yk, n ∈ N, k=1 is called a random walk. -4 0 8 16 Random walks have stationary and independent increments Yk = Sk − Sk−1, k ≥ 1. Stationarity means the Yk have identical distribution. Definition 2 A right-continuous process Xt, t ∈ R+, with stationary independent increments is called L´evy process. 2 Page 1 What are Sn, n ≥ 0, and Xt, t ≥ 0? Stochastic processes; mathematical objects, well-defined, with many nice properties that can be studied. If you don’t like this, think of a model for a stock price evolving with time. There are also many other applications. If you worry about negative values, think of log’s of prices. What does Definition 2 mean? Increments , = 1 , are independent and Xtk − Xtk−1 k , . , n , = 1 for all 0 = . Xtk − Xtk−1 ∼ Xtk−tk−1 k , . , n t0 < . < tn Right-continuity refers to the sample paths (realisations). 3 Can we obtain L´evyprocesses from random walks? What happens e.g. if we let the time unit tend to zero, i.e. take a more and more remote look at our random walk? If we focus at a fixed time, 1 say, and speed up the process so as to make n steps per time unit, we know what happens, the answer is given by the Central Limit Theorem: 2 Theorem 1 (Lindeberg-L´evy) If σ = V ar(Y1) < ∞, then Sn − (Sn) √E → Z ∼ N(0, σ2) in distribution, as n → ∞. -
Spatio-Temporal Cluster Detection and Local Moran Statistics of Point Processes
Old Dominion University ODU Digital Commons Mathematics & Statistics Theses & Dissertations Mathematics & Statistics Spring 2019 Spatio-Temporal Cluster Detection and Local Moran Statistics of Point Processes Jennifer L. Matthews Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/mathstat_etds Part of the Applied Statistics Commons, and the Biostatistics Commons Recommended Citation Matthews, Jennifer L.. "Spatio-Temporal Cluster Detection and Local Moran Statistics of Point Processes" (2019). Doctor of Philosophy (PhD), Dissertation, Mathematics & Statistics, Old Dominion University, DOI: 10.25777/3mps-rk62 https://digitalcommons.odu.edu/mathstat_etds/46 This Dissertation is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion in Mathematics & Statistics Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. ABSTRACT Approved for public release; distribution is unlimited SPATIO-TEMPORAL CLUSTER DETECTION AND LOCAL MORAN STATISTICS OF POINT PROCESSES Jennifer L. Matthews Commander, United States Navy Old Dominion University, 2019 Director: Dr. Norou Diawara Moran's index is a statistic that measures spatial dependence, quantifying the degree of dispersion or clustering of point processes and events in some location/area. Recognizing that a single Moran's index may not give a sufficient summary of the spatial autocorrelation measure, a local -
The Concentration of Measure Phenomenon
http://dx.doi.org/10.1090/surv/089 Selected Titles in This Series 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2001 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, 2000 80 Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, 2000 79 Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz, The semicircle law, free random variables and entropy, 2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, 2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, 1999 69 C. -
Non-Local Branching Superprocesses and Some Related Models
Published in: Acta Applicandae Mathematicae 74 (2002), 93–112. Non-local Branching Superprocesses and Some Related Models Donald A. Dawson1 School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Canada K1S 5B6 E-mail: [email protected] Luis G. Gorostiza2 Departamento de Matem´aticas, Centro de Investigaci´ony de Estudios Avanzados, A.P. 14-740, 07000 M´exicoD. F., M´exico E-mail: [email protected] Zenghu Li3 Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China E-mail: [email protected] Abstract A new formulation of non-local branching superprocesses is given from which we derive as special cases the rebirth, the multitype, the mass- structured, the multilevel and the age-reproduction-structured superpro- cesses and the superprocess-controlled immigration process. This unified treatment simplifies considerably the proof of existence of the old classes of superprocesses and also gives rise to some new ones. AMS Subject Classifications: 60G57, 60J80 Key words and phrases: superprocess, non-local branching, rebirth, mul- titype, mass-structured, multilevel, age-reproduction-structured, superprocess- controlled immigration. 1Supported by an NSERC Research Grant and a Max Planck Award. 2Supported by the CONACYT (Mexico, Grant No. 37130-E). 3Supported by the NNSF (China, Grant No. 10131040). 1 1 Introduction Measure-valued branching processes or superprocesses constitute a rich class of infinite dimensional processes currently under rapid development. Such processes arose in appli- cations as high density limits of branching particle systems; see e.g. Dawson (1992, 1993), Dynkin (1993, 1994), Watanabe (1968). The development of this subject has been stimu- lated from different subjects including branching processes, interacting particle systems, stochastic partial differential equations and non-linear partial differential equations. -
Simulation of Markov Chains
Copyright c 2007 by Karl Sigman 1 Simulating Markov chains Many stochastic processes used for the modeling of financial assets and other systems in engi- neering are Markovian, and this makes it relatively easy to simulate from them. Here we present a brief introduction to the simulation of Markov chains. Our emphasis is on discrete-state chains both in discrete and continuous time, but some examples with a general state space will be discussed too. 1.1 Definition of a Markov chain We shall assume that the state space S of our Markov chain is S = ZZ= f:::; −2; −1; 0; 1; 2;:::g, the integers, or a proper subset of the integers. Typical examples are S = IN = f0; 1; 2 :::g, the non-negative integers, or S = f0; 1; 2 : : : ; ag, or S = {−b; : : : ; 0; 1; 2 : : : ; ag for some integers a; b > 0, in which case the state space is finite. Definition 1.1 A stochastic process fXn : n ≥ 0g is called a Markov chain if for all times n ≥ 0 and all states i0; : : : ; i; j 2 S, P (Xn+1 = jjXn = i; Xn−1 = in−1;:::;X0 = i0) = P (Xn+1 = jjXn = i) (1) = Pij: Pij denotes the probability that the chain, whenever in state i, moves next (one unit of time later) into state j, and is referred to as a one-step transition probability. The square matrix P = (Pij); i; j 2 S; is called the one-step transition matrix, and since when leaving state i the chain must move to one of the states j 2 S, each row sums to one (e.g., forms a probability distribution): For each i X Pij = 1: j2S We are assuming that the transition probabilities do not depend on the time n, and so, in particular, using n = 0 in (1) yields Pij = P (X1 = jjX0 = i): (Formally we are considering only time homogenous MC's meaning that their transition prob- abilities are time-homogenous (time stationary).) The defining property (1) can be described in words as the future is independent of the past given the present state. -
A Representation for Functionals of Superprocesses Via Particle Picture Raisa E
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector stochastic processes and their applications ELSEVIER Stochastic Processes and their Applications 64 (1996) 173-186 A representation for functionals of superprocesses via particle picture Raisa E. Feldman *,‘, Srikanth K. Iyer ‘,* Department of Statistics and Applied Probability, University oj’ Cull~ornia at Santa Barbara, CA 93/06-31/O, USA, and Department of Industrial Enyineeriny and Management. Technion - Israel Institute of’ Technology, Israel Received October 1995; revised May 1996 Abstract A superprocess is a measure valued process arising as the limiting density of an infinite col- lection of particles undergoing branching and diffusion. It can also be defined as a measure valued Markov process with a specified semigroup. Using the latter definition and explicit mo- ment calculations, Dynkin (1988) built multiple integrals for the superprocess. We show that the multiple integrals of the superprocess defined by Dynkin arise as weak limits of linear additive functionals built on the particle system. Keywords: Superprocesses; Additive functionals; Particle system; Multiple integrals; Intersection local time AMS c.lassijication: primary 60555; 60H05; 60580; secondary 60F05; 6OG57; 60Fl7 1. Introduction We study a class of functionals of a superprocess that can be identified as the multiple Wiener-It6 integrals of this measure valued process. More precisely, our objective is to study these via the underlying branching particle system, thus providing means of thinking about the functionals in terms of more simple, intuitive and visualizable objects. This way of looking at multiple integrals of a stochastic process has its roots in the previous work of Adler and Epstein (=Feldman) (1987), Epstein (1989), Adler ( 1989) Feldman and Rachev (1993), Feldman and Krishnakumar ( 1994). -
Laplace Transform Identities for the Volume of Stopping Sets Based on Poisson Point Processes
Laplace transform identities for the volume of stopping sets based on Poisson point processes Nicolas Privault Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University 21 Nanyang Link Singapore 637371 November 9, 2015 Abstract We derive Laplace transform identities for the volume content of random stopping sets based on Poisson point processes. Our results are based on antic- ipating Girsanov identities for Poisson point processes under a cyclic vanishing condition for a finite difference gradient. This approach does not require classi- cal assumptions based on set-indexed martingales and the (partial) ordering of index sets. The examples treated focus on stopping sets in finite volume, and include the random missed volume of Poisson convex hulls. Key words: Poisson point processes; stopping sets; gamma-type distributions; Gir- sanov identities; anticipating stochastic calculus. Mathematics Subject Classification (2010): 60D05; 60G40; 60G57; 60G48; 60H07. 1 Introduction Gamma-type results for the area of random domains constructed from a finite number of \typical" Poisson distributed points, and more generally known as complementary theorems, have been obtained in [7], cf. e.g. Theorem 10.4.8 in [14]. 1 Stopping sets are random sets that carry over the notion of stopping time to set- indexed processes, cf. Definition 2.27 in [8], based on stochastic calculus for set- indexed martingales, cf. e.g. [6]. Gamma-type results for the probability law of the volume content of random sets have been obtained in the framework of stopping sets in [15], via Laplace transforms, using the martingale property of set-indexed stochas- tic exponentials, see [16] for the strong Markov property for point processes, cf.