Stochastic Processes (Week 6)
|
|
- Camilla McGee
- 5 years ago
- Views:
Transcription
1 Stochastic Processes (Week 6) October 30th, Discrete-time Finite Markov Chains 2 Countable Markov Chains 3 Continuous-Time Markov Chains 3.1 Poisson Process 3.2 Finite State Space Kolmogrov s backward and forward equation Kolmogorov s forward and backward equation, Embedded Markov chain Large time behavior Definition 3.1. A continuous-time Markov chain is irreducible if all states communicate, i.e, for each x, y S, there exists a sequence of z 1, z 2,, z j S with α(x, z 1 ), α(z 1, z 2 ),, α(z j 1, z j ), α(z j, y) all strictly positive. About periodicity, it does not occur to continuous-time Markov chain because Lemma 3.1. For any irreducible continuous-time Markov chain, P t has strictly positive entries for all t > 0. Proof. (i). if x = y, it is trivial to check that for any integer n, P t (x, x) Pt/n n (x, x), if P t(x, x) = 0, then P t/n (x, x) = 0 for any integer n, it contradicts with P 0 (x, x) = 1; 1
2 (ii). if x y, by definition, there exists k 1, k 2,, k m such that P t/2 n(x, k 1 )P t/2 n(k 1, k 2 ) P t/2 n(k m 1, k m )P t/2 n(k m, y) lim = n + (t/2 n ) m+1 = α(x, k 1 )α(k 1, k 2 ) α(k m 1, k m )α(k m, y) > 0. That is, P t/2 n(x, k 1 )P t/2 n(k 1, k 2 ) P t/2 n(k m 1, k m )P t/2 n(k m, y) > 0 for sufficiently large n. Therefore, P t (x, y) P t/2 n(x, k 1 )P t/2 n(k 1, k 2 ) P t/2 n(k m, y)p t (m+1)/2 n t(y, y) > 0. Remark: From the proof we can conclude that, an equivalent definition for irreducibility is, a continuous-time Markov chain is irreducible if for every x, y S, P t (x, y) > 0 for some t. Corollary 3.1. A continuous time Markov chain is reducible if and only if its embedded chain is irreducible. Denote H x as the holding time in state x, i.e, H x = inf{t > 0 : X t x, X 0 = x}, and T x,x = inf{t H x : X t = x, X 0 = x}, the amount of time until the Markov chain re-visits state x after the first change of state if X 0 = x. Definition 3.2. State x is called recurrent if with probability 1, the Markov chain will return to state x within a finite interval of time, i,e, P (T x,x < ) = 1. Otherwise, it is called transient. Remark: A state x for a continuous time Markov chain is recurrent/transient if and only if it is recurrent/transient for the embedded discrete-time chain. Consequently, an irreducible Markov continuous time Markov chain is recurrent. Moreover, if it is recurrent, the total amount of time that X t stays at x, 0 I(X s = x)ds is infinite with probability 1. Proof. Note that X 0 = Y 0 = x, define τ x = inf{n 1 : Y n = x}, then T x,x = T 1 + T T τx. T x,x = if and only if τ x =. I(X 0 s = x)ds = Nx H x,k where H x,k s are independent exponential distributed random variables with rate α(x) and N x = lim I(Y n = x). n=0 k=1 2
3 Definition 3.3. For continuous time Markov chain, is said to be an invariant probability distribution if P t = for all t > 0. Lemma 3.2. A nonnegative vector with 1 = 1 is an invariant probability distribution if and only if A = 0. Proof. If P t = for all t > 0, then 0 = d(p t)(y) = π(x) dp t(x, y) = π(x) p t (x, z)a z,y z S = π(x)p t (x, z)a z,y = π(z)a z,y = (A)(y). z S z S Conversely, if A = 0, then ( ) d π(x)p t (x, y) = π(x) dp t(x, y) = π(x) A xz P t (z, y) z S = π(x)a xz P t (z, y) = (A)(z)P t (z, y) = 0. z S z S P t is constant and P t = P 0 =. Note that for an irreducible continuous-time Markov chain X t with finite state space S, the embedded discrete time Markov chain Y n is irreducible, recurrent with finite state space S, hence there exists a unique positive invariant probability vector. Denote the one-step transition matrix of Y n is P. Since A x,y = α(x)p (x, y) for x y, and α(x) for x = y, Through direct calculation, for any η = (η(x)), we have η τ A = 0 η(x)a x,y = 0 y S x y η(x)α(x)p (x, y) = η(y)α(y) P = where = (π(x)), π(x) = η(x)α(x). i.e, this is proportional to the unique invariant probability vector of Y n. Hence, there exists a unique positive probability vector η satisfying η τ A = 0 and η(x) = ( ) 1 π(x) π(x) α(x) α(x). Another method of verification is to follow exercise 3.4 in the textbook: 3
4 Theorem 3.1. For an irreducible continuous-time Markov chain with finite s- tate space S, there is a unique probability vector satisfying A = 0; all the eigenvalues of A have negative real part. Proof. Note that A is the infinitesimal generator for an irreducible continuoustime Markov chain, it has such properties: the row sums equal to 0; diagonal elements are nonnegative; off-diagonal elements are nonnegative. Let a be some positive number greater than the absolute values of all the entries of A, then P = 1 a A + I is the transition matrix for a discrete-time, irreducible, aperiodic Markov Chain: (i). P xy = 1 a A xy + I(x = y) 0; (ii). y P xy = 1 A xy + 1 = 1; a y (iii). P xx = 1 a A xx + 1 > 0; (iv). by definition, for each x, y S, there exists j distinct z 1, z 2,, z j with A xz1, A z1 z 2,, A zj 1 z j, A zj y are strictly positive. S P j+1 xy i.e, x y. P xz1 P z1 z 2 P zj 1 z j P zj y A xz 1 A z1 z 2 A zj 1 z j A zj y a j+1 > 0, For an irreducible, aperiodic Markov chain, by Perron-Frobenius Theorem, P has a unique left eigenvector with eigenvalue 1 and that is a probability vector, all the other eigenvalues of P have absolute values strictly less than 1. Theorem 3.2. For an irreducible continuous-time Markov chain with finite state space S, lim P t =, where A = 0. t + Proof. Fixed t > 0, Q = P t can be considered as the one-step transition probability matrix for an irreducible, aperiodic discrete-time Markov chain with finite state space S, lim Q l = where is the invariant probability distribution l of the Markov chain, it is independent of choice of t > 0. Moreover, A dp t 0 = lim t = lim P ta =. t A 4
5 3.2.3 Exit distributions and hitting times Suppose X t is a continuous time irreducible Markov chain on finite state space S. I. Define T = inf{t 0 : X t x}, i.e, the time of the first exist from x then T is exponential with parameter α(x) if assume X 0 = x, therefore, E(T X 0 = x) = 1/α(x). II. For a fixed state z S, define Y = inf{t 0 : X t = z}, i.e, the time of the first visit to z Define b(x) = E(Y X 0 = x),clearly, b(z) = 0. Denote b = (b(x)) x z. Theorem 3.3. Let à be the matrix obtained from A by deleting the row and the column associated to the state z, then b = à 1 1. Lemma 3.3. The à in Theorem 3.3 is invertible. Proof. We observe that for Ã, the row sums are all nonpositive and at least one of these row sums is strictly negative. Otherwise, α(z, y) = 0 for all y z, contradicts with the irreducibility assumption. From Theorem 3.1, we know that ( πa ) = 0, 1 T π = 1 has a unique positive A T solution. Suppose S = n, then rank = n, and rank(a) = n 1, hence {π : πa = 0} has dimension 1. Denote the adjoint matrix of A by A, then A A = 0, each row of A is a solution to πa = 0 and all entries are nonzero, therefore, à is invertible. Proof of Theorem 3.3. By definition, for x z, b(x) = E(Y X 0 = x) = E(T X 0 = x) + = 1 α(x) + i.e, α(x)b(x) = 1 + y S,y x = 1 + y S,y x y S,y x,z 1 T α(x, y) α(x) b(y), α(x, y)b(y) α(x, y)b(y), 5 y S,y x P (X T = y X 0 = x)e(y X 0 = y)
6 which implies 1 + y z A x,y b(y) = 0, i.e, 0 = 1 + Ã b, b = Ã 1 1. Example 3.1. Consider a Markov chain with four states {0, 1, 2, 3}, and infinitesimal generator, A = let z = 3, compute b = (b(x)) x 3 where b(x) = E(Y X 0 = x), Y = inf{t 0 : X t = 3}. Exercises: Ã = , b = Ã 1 1 = (8/3, 5/3, 4/3) Assume = (π(x)) is the invariant probability distribution for an irreducible continuous time Markov chain, let V x (t) = t I(X 0 s = x)ds, i.e, the time spent in state x up to time t, show that V x (t) lim t t = π(x) almost surely, i.e, π(x) is the proportion of time spent in state x over long periods of time. 2. (Detailed balance condition) Please show that (1) For a discrete-time Markov chain with one-step transition probability matrix P and state space S, if a nonnegative vector = (π(x)) satisfies π(x)p(x, y) = π(y)p(y, x) for all x y and π(x) = 1, then is an invariant probability distribution. (2) For a continuous-time Markov chain with infinitesimal generator A and state space S, if a nonnegative vector = (π(x)) satisfies π(x)a x,y = π(y)a y,x for all x y and π(x) = 1, then is an invariant probability distribution. 6
MATH 56A: STOCHASTIC PROCESSES CHAPTER 2
MATH 56A: STOCHASTIC PROCESSES CHAPTER 2 2. Countable Markov Chains I started Chapter 2 which talks about Markov chains with a countably infinite number of states. I did my favorite example which is on
More informationChapter 7. Markov chain background. 7.1 Finite state space
Chapter 7 Markov chain background A stochastic process is a family of random variables {X t } indexed by a varaible t which we will think of as time. Time can be discrete or continuous. We will only consider
More informationMarkov Chains, Stochastic Processes, and Matrix Decompositions
Markov Chains, Stochastic Processes, and Matrix Decompositions 5 May 2014 Outline 1 Markov Chains Outline 1 Markov Chains 2 Introduction Perron-Frobenius Matrix Decompositions and Markov Chains Spectral
More informationMATH 56A: STOCHASTIC PROCESSES CHAPTER 7
MATH 56A: STOCHASTIC PROCESSES CHAPTER 7 7. Reversal This chapter talks about time reversal. A Markov process is a state X t which changes with time. If we run time backwards what does it look like? 7.1.
More informationStochastic Processes
Stochastic Processes 8.445 MIT, fall 20 Mid Term Exam Solutions October 27, 20 Your Name: Alberto De Sole Exercise Max Grade Grade 5 5 2 5 5 3 5 5 4 5 5 5 5 5 6 5 5 Total 30 30 Problem :. True / False
More informationMATH 56A SPRING 2008 STOCHASTIC PROCESSES 65
MATH 56A SPRING 2008 STOCHASTIC PROCESSES 65 2.2.5. proof of extinction lemma. The proof of Lemma 2.3 is just like the proof of the lemma I did on Wednesday. It goes like this. Suppose that â is the smallest
More informationPositive and null recurrent-branching Process
December 15, 2011 In last discussion we studied the transience and recurrence of Markov chains There are 2 other closely related issues about Markov chains that we address Is there an invariant distribution?
More informationLecture 5: Random Walks and Markov Chain
Spectral Graph Theory and Applications WS 20/202 Lecture 5: Random Walks and Markov Chain Lecturer: Thomas Sauerwald & He Sun Introduction to Markov Chains Definition 5.. A sequence of random variables
More informationNote that in the example in Lecture 1, the state Home is recurrent (and even absorbing), but all other states are transient. f ii (n) f ii = n=1 < +
Random Walks: WEEK 2 Recurrence and transience Consider the event {X n = i for some n > 0} by which we mean {X = i}or{x 2 = i,x i}or{x 3 = i,x 2 i,x i},. Definition.. A state i S is recurrent if P(X n
More informationLecture 7. We can regard (p(i, j)) as defining a (maybe infinite) matrix P. Then a basic fact is
MARKOV CHAINS What I will talk about in class is pretty close to Durrett Chapter 5 sections 1-5. We stick to the countable state case, except where otherwise mentioned. Lecture 7. We can regard (p(i, j))
More informationStatistics 150: Spring 2007
Statistics 150: Spring 2007 April 23, 2008 0-1 1 Limiting Probabilities If the discrete-time Markov chain with transition probabilities p ij is irreducible and positive recurrent; then the limiting probabilities
More information8. Statistical Equilibrium and Classification of States: Discrete Time Markov Chains
8. Statistical Equilibrium and Classification of States: Discrete Time Markov Chains 8.1 Review 8.2 Statistical Equilibrium 8.3 Two-State Markov Chain 8.4 Existence of P ( ) 8.5 Classification of States
More informationProbability & Computing
Probability & Computing Stochastic Process time t {X t t 2 T } state space Ω X t 2 state x 2 discrete time: T is countable T = {0,, 2,...} discrete space: Ω is finite or countably infinite X 0,X,X 2,...
More informationMATH 56A: STOCHASTIC PROCESSES CHAPTER 1
MATH 56A: STOCHASTIC PROCESSES CHAPTER. Finite Markov chains For the sake of completeness of these notes I decided to write a summary of the basic concepts of finite Markov chains. The topics in this chapter
More informationA review of Continuous Time MC STA 624, Spring 2015
A review of Continuous Time MC STA 624, Spring 2015 Ruriko Yoshida Dept. of Statistics University of Kentucky polytopes.net STA 624 1 Continuous Time Markov chains Definition A continuous time stochastic
More informationP i [B k ] = lim. n=1 p(n) ii <. n=1. V i :=
2.7. Recurrence and transience Consider a Markov chain {X n : n N 0 } on state space E with transition matrix P. Definition 2.7.1. A state i E is called recurrent if P i [X n = i for infinitely many n]
More informationSTOCHASTIC PROCESSES Basic notions
J. Virtamo 38.3143 Queueing Theory / Stochastic processes 1 STOCHASTIC PROCESSES Basic notions Often the systems we consider evolve in time and we are interested in their dynamic behaviour, usually involving
More informationMarkov processes and queueing networks
Inria September 22, 2015 Outline Poisson processes Markov jump processes Some queueing networks The Poisson distribution (Siméon-Denis Poisson, 1781-1840) { } e λ λ n n! As prevalent as Gaussian distribution
More informationChapter 2. Markov Chains. Introduction
Chapter 2 Markov Chains Introduction A Markov chain is a sequence of random variables {X n ; n = 0, 1, 2,...}, defined on some probability space (Ω, F, IP), taking its values in a set E which could be
More informationMarkov chains. Randomness and Computation. Markov chains. Markov processes
Markov chains Randomness and Computation or, Randomized Algorithms Mary Cryan School of Informatics University of Edinburgh Definition (Definition 7) A discrete-time stochastic process on the state space
More informationMarkov Chains and Stochastic Sampling
Part I Markov Chains and Stochastic Sampling 1 Markov Chains and Random Walks on Graphs 1.1 Structure of Finite Markov Chains We shall only consider Markov chains with a finite, but usually very large,
More informationMath 456: Mathematical Modeling. Tuesday, March 6th, 2018
Math 456: Mathematical Modeling Tuesday, March 6th, 2018 Markov Chains: Exit distributions and the Strong Markov Property Tuesday, March 6th, 2018 Last time 1. Weighted graphs. 2. Existence of stationary
More informationMARKOV CHAINS AND HIDDEN MARKOV MODELS
MARKOV CHAINS AND HIDDEN MARKOV MODELS MERYL SEAH Abstract. This is an expository paper outlining the basics of Markov chains. We start the paper by explaining what a finite Markov chain is. Then we describe
More informationMarkov chains. 1 Discrete time Markov chains. c A. J. Ganesh, University of Bristol, 2015
Markov chains c A. J. Ganesh, University of Bristol, 2015 1 Discrete time Markov chains Example: A drunkard is walking home from the pub. There are n lampposts between the pub and his home, at each of
More informationModern Discrete Probability Spectral Techniques
Modern Discrete Probability VI - Spectral Techniques Background Sébastien Roch UW Madison Mathematics December 22, 2014 1 Review 2 3 4 Mixing time I Theorem (Convergence to stationarity) Consider a finite
More informationLecture 7. µ(x)f(x). When µ is a probability measure, we say µ is a stationary distribution.
Lecture 7 1 Stationary measures of a Markov chain We now study the long time behavior of a Markov Chain: in particular, the existence and uniqueness of stationary measures, and the convergence of the distribution
More informationMS&E 321 Spring Stochastic Systems June 1, 2013 Prof. Peter W. Glynn Page 1 of 10. x n+1 = f(x n ),
MS&E 321 Spring 12-13 Stochastic Systems June 1, 2013 Prof. Peter W. Glynn Page 1 of 10 Section 4: Steady-State Theory Contents 4.1 The Concept of Stochastic Equilibrium.......................... 1 4.2
More informationLecture 5. If we interpret the index n 0 as time, then a Markov chain simply requires that the future depends only on the present and not on the past.
1 Markov chain: definition Lecture 5 Definition 1.1 Markov chain] A sequence of random variables (X n ) n 0 taking values in a measurable state space (S, S) is called a (discrete time) Markov chain, if
More informationCS145: Probability & Computing Lecture 18: Discrete Markov Chains, Equilibrium Distributions
CS145: Probability & Computing Lecture 18: Discrete Markov Chains, Equilibrium Distributions Instructor: Erik Sudderth Brown University Computer Science April 14, 215 Review: Discrete Markov Chains Some
More informationSMSTC (2007/08) Probability.
SMSTC (27/8) Probability www.smstc.ac.uk Contents 12 Markov chains in continuous time 12 1 12.1 Markov property and the Kolmogorov equations.................... 12 2 12.1.1 Finite state space.................................
More informationSummary of Stochastic Processes
Summary of Stochastic Processes Kui Tang May 213 Based on Lawler s Introduction to Stochastic Processes, second edition, and course slides from Prof. Hongzhong Zhang. Contents 1 Difference/tial Equations
More informationApplied Stochastic Processes
Applied Stochastic Processes Jochen Geiger last update: July 18, 2007) Contents 1 Discrete Markov chains........................................ 1 1.1 Basic properties and examples................................
More informationSTATS 3U03. Sang Woo Park. March 29, Textbook: Inroduction to stochastic processes. Requirement: 5 assignments, 2 tests, and 1 final
STATS 3U03 Sang Woo Park March 29, 2017 Course Outline Textbook: Inroduction to stochastic processes Requirement: 5 assignments, 2 tests, and 1 final Test 1: Friday, February 10th Test 2: Friday, March
More informationMarkov Chains on Countable State Space
Markov Chains on Countable State Space 1 Markov Chains Introduction 1. Consider a discrete time Markov chain {X i, i = 1, 2,...} that takes values on a countable (finite or infinite) set S = {x 1, x 2,...},
More informationNecessary and sufficient conditions for strong R-positivity
Necessary and sufficient conditions for strong R-positivity Wednesday, November 29th, 2017 The Perron-Frobenius theorem Let A = (A(x, y)) x,y S be a nonnegative matrix indexed by a countable set S. We
More informationhttp://www.math.uah.edu/stat/markov/.xhtml 1 of 9 7/16/2009 7:20 AM Virtual Laboratories > 16. Markov Chains > 1 2 3 4 5 6 7 8 9 10 11 12 1. A Markov process is a random process in which the future is
More informationMarkov Chains. X(t) is a Markov Process if, for arbitrary times t 1 < t 2 <... < t k < t k+1. If X(t) is discrete-valued. If X(t) is continuous-valued
Markov Chains X(t) is a Markov Process if, for arbitrary times t 1 < t 2
More informationMarkov Chains, Random Walks on Graphs, and the Laplacian
Markov Chains, Random Walks on Graphs, and the Laplacian CMPSCI 791BB: Advanced ML Sridhar Mahadevan Random Walks! There is significant interest in the problem of random walks! Markov chain analysis! Computer
More informationClassification of Countable State Markov Chains
Classification of Countable State Markov Chains Friday, March 21, 2014 2:01 PM How can we determine whether a communication class in a countable state Markov chain is: transient null recurrent positive
More informationConvex Optimization CMU-10725
Convex Optimization CMU-10725 Simulated Annealing Barnabás Póczos & Ryan Tibshirani Andrey Markov Markov Chains 2 Markov Chains Markov chain: Homogen Markov chain: 3 Markov Chains Assume that the state
More informationDiscrete time Markov chains. Discrete Time Markov Chains, Limiting. Limiting Distribution and Classification. Regular Transition Probability Matrices
Discrete time Markov chains Discrete Time Markov Chains, Limiting Distribution and Classification DTU Informatics 02407 Stochastic Processes 3, September 9 207 Today: Discrete time Markov chains - invariant
More information12 Markov chains The Markov property
12 Markov chains Summary. The chapter begins with an introduction to discrete-time Markov chains, and to the use of matrix products and linear algebra in their study. The concepts of recurrence and transience
More informationMATH36001 Perron Frobenius Theory 2015
MATH361 Perron Frobenius Theory 215 In addition to saying something useful, the Perron Frobenius theory is elegant. It is a testament to the fact that beautiful mathematics eventually tends to be useful,
More informationStochastic Simulation
Stochastic Simulation Ulm University Institute of Stochastics Lecture Notes Dr. Tim Brereton Summer Term 2015 Ulm, 2015 2 Contents 1 Discrete-Time Markov Chains 5 1.1 Discrete-Time Markov Chains.....................
More informationAn Introduction to Entropy and Subshifts of. Finite Type
An Introduction to Entropy and Subshifts of Finite Type Abby Pekoske Department of Mathematics Oregon State University pekoskea@math.oregonstate.edu August 4, 2015 Abstract This work gives an overview
More informationRECURRENCE IN COUNTABLE STATE MARKOV CHAINS
RECURRENCE IN COUNTABLE STATE MARKOV CHAINS JIN WOO SUNG Abstract. This paper investigates the recurrence and transience of countable state irreducible Markov chains. Recurrence is the property that a
More informationLecture #5. Dependencies along the genome
Markov Chains Lecture #5 Background Readings: Durbin et. al. Section 3., Polanski&Kimmel Section 2.8. Prepared by Shlomo Moran, based on Danny Geiger s and Nir Friedman s. Dependencies along the genome
More informationLecture 6. 2 Recurrence/transience, harmonic functions and martingales
Lecture 6 Classification of states We have shown that all states of an irreducible countable state Markov chain must of the same tye. This gives rise to the following classification. Definition. [Classification
More informationSTAT STOCHASTIC PROCESSES. Contents
STAT 3911 - STOCHASTIC PROCESSES ANDREW TULLOCH Contents 1. Stochastic Processes 2 2. Classification of states 2 3. Limit theorems for Markov chains 4 4. First step analysis 5 5. Branching processes 5
More informationStatistics 992 Continuous-time Markov Chains Spring 2004
Summary Continuous-time finite-state-space Markov chains are stochastic processes that are widely used to model the process of nucleotide substitution. This chapter aims to present much of the mathematics
More informationHomework set 3 - Solutions
Homework set 3 - Solutions Math 495 Renato Feres Problems 1. (Text, Exercise 1.13, page 38.) Consider the Markov chain described in Exercise 1.1: The Smiths receive the paper every morning and place it
More informationTheory and Applications of Stochastic Systems Lecture Exponential Martingale for Random Walk
Instructor: Victor F. Araman December 4, 2003 Theory and Applications of Stochastic Systems Lecture 0 B60.432.0 Exponential Martingale for Random Walk Let (S n : n 0) be a random walk with i.i.d. increments
More informationThe Markov Chain Monte Carlo Method
The Markov Chain Monte Carlo Method Idea: define an ergodic Markov chain whose stationary distribution is the desired probability distribution. Let X 0, X 1, X 2,..., X n be the run of the chain. The Markov
More informationLecture 11: Introduction to Markov Chains. Copyright G. Caire (Sample Lectures) 321
Lecture 11: Introduction to Markov Chains Copyright G. Caire (Sample Lectures) 321 Discrete-time random processes A sequence of RVs indexed by a variable n 2 {0, 1, 2,...} forms a discretetime random process
More informationTransience: Whereas a finite closed communication class must be recurrent, an infinite closed communication class can be transient:
Stochastic2010 Page 1 Long-Time Properties of Countable-State Markov Chains Tuesday, March 23, 2010 2:14 PM Homework 2: if you turn it in by 5 PM on 03/25, I'll grade it by 03/26, but you can turn it in
More informationZdzis law Brzeźniak and Tomasz Zastawniak
Basic Stochastic Processes by Zdzis law Brzeźniak and Tomasz Zastawniak Springer-Verlag, London 1999 Corrections in the 2nd printing Version: 21 May 2005 Page and line numbers refer to the 2nd printing
More informationLecture 2: September 8
CS294 Markov Chain Monte Carlo: Foundations & Applications Fall 2009 Lecture 2: September 8 Lecturer: Prof. Alistair Sinclair Scribes: Anand Bhaskar and Anindya De Disclaimer: These notes have not been
More informationContinuous Time Markov Chain Examples
Continuous Markov Chain Examples Example Consider a continuous time Markov chain on S {,, } The Markov chain is a model that describes the current status of a match between two particular contestants:
More informationMarkov processes Course note 2. Martingale problems, recurrence properties of discrete time chains.
Institute for Applied Mathematics WS17/18 Massimiliano Gubinelli Markov processes Course note 2. Martingale problems, recurrence properties of discrete time chains. [version 1, 2017.11.1] We introduce
More information2. Transience and Recurrence
Virtual Laboratories > 15. Markov Chains > 1 2 3 4 5 6 7 8 9 10 11 12 2. Transience and Recurrence The study of Markov chains, particularly the limiting behavior, depends critically on the random times
More information1 Continuous-time chains, finite state space
Université Paris Diderot 208 Markov chains Exercises 3 Continuous-time chains, finite state space Exercise Consider a continuous-time taking values in {, 2, 3}, with generator 2 2. 2 2 0. Draw the diagramm
More information2 Discrete-Time Markov Chains
2 Discrete-Time Markov Chains Angela Peace Biomathematics II MATH 5355 Spring 2017 Lecture notes follow: Allen, Linda JS. An introduction to stochastic processes with applications to biology. CRC Press,
More informationSpectral radius, symmetric and positive matrices
Spectral radius, symmetric and positive matrices Zdeněk Dvořák April 28, 2016 1 Spectral radius Definition 1. The spectral radius of a square matrix A is ρ(a) = max{ λ : λ is an eigenvalue of A}. For an
More information1 Invariant subspaces
MATH 2040 Linear Algebra II Lecture Notes by Martin Li Lecture 8 Eigenvalues, eigenvectors and invariant subspaces 1 In previous lectures we have studied linear maps T : V W from a vector space V to another
More informationDefinition A finite Markov chain is a memoryless homogeneous discrete stochastic process with a finite number of states.
Chapter 8 Finite Markov Chains A discrete system is characterized by a set V of states and transitions between the states. V is referred to as the state space. We think of the transitions as occurring
More informationLECTURE NOTES ELEMENTARY NUMERICAL METHODS. Eusebius Doedel
LECTURE NOTES on ELEMENTARY NUMERICAL METHODS Eusebius Doedel TABLE OF CONTENTS Vector and Matrix Norms 1 Banach Lemma 20 The Numerical Solution of Linear Systems 25 Gauss Elimination 25 Operation Count
More informationMath Homework 5 Solutions
Math 45 - Homework 5 Solutions. Exercise.3., textbook. The stochastic matrix for the gambler problem has the following form, where the states are ordered as (,, 4, 6, 8, ): P = The corresponding diagram
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
More informationStochastic Processes MIT, fall 2011 Day by day lecture outline and weekly homeworks. A) Lecture Outline Suggested reading
Stochastic Processes 18445 MIT, fall 2011 Day by day lecture outline and weekly homeworks A) Lecture Outline Suggested reading Part 1: Random walk on Z Lecture 1: thursday, september 8, 2011 Presentation
More informationPerron Frobenius Theory
Perron Frobenius Theory Oskar Perron Georg Frobenius (1880 1975) (1849 1917) Stefan Güttel Perron Frobenius Theory 1 / 10 Positive and Nonnegative Matrices Let A, B R m n. A B if a ij b ij i, j, A > B
More informationNon-Essential Uses of Probability in Analysis Part IV Efficient Markovian Couplings. Krzysztof Burdzy University of Washington
Non-Essential Uses of Probability in Analysis Part IV Efficient Markovian Couplings Krzysztof Burdzy University of Washington 1 Review See B and Kendall (2000) for more details. See also the unpublished
More informationSTA 624 Practice Exam 2 Applied Stochastic Processes Spring, 2008
Name STA 624 Practice Exam 2 Applied Stochastic Processes Spring, 2008 There are five questions on this test. DO use calculators if you need them. And then a miracle occurs is not a valid answer. There
More informationMATH 56A: STOCHASTIC PROCESSES CHAPTER 3
MATH 56A: STOCHASTIC PROCESSES CHAPTER 3 Plan for rest of semester (1) st week (8/31, 9/6, 9/7) Chap 0: Diff eq s an linear recursion (2) n week (9/11...) Chap 1: Finite Markov chains (3) r week (9/18...)
More informationMarkov Chains CK eqns Classes Hitting times Rec./trans. Strong Markov Stat. distr. Reversibility * Markov Chains
Markov Chains A random process X is a family {X t : t T } of random variables indexed by some set T. When T = {0, 1, 2,... } one speaks about a discrete-time process, for T = R or T = [0, ) one has a continuous-time
More informationCONVERGENCE THEOREM FOR FINITE MARKOV CHAINS. Contents
CONVERGENCE THEOREM FOR FINITE MARKOV CHAINS ARI FREEDMAN Abstract. In this expository paper, I will give an overview of the necessary conditions for convergence in Markov chains on finite state spaces.
More information6 Markov Chain Monte Carlo (MCMC)
6 Markov Chain Monte Carlo (MCMC) The underlying idea in MCMC is to replace the iid samples of basic MC methods, with dependent samples from an ergodic Markov chain, whose limiting (stationary) distribution
More informationarxiv: v2 [math.pr] 25 May 2017
STATISTICAL ANALYSIS OF THE FIRST PASSAGE PATH ENSEMBLE OF JUMP PROCESSES MAX VON KLEIST, CHRISTOF SCHÜTTE, 2, AND WEI ZHANG arxiv:70.04270v2 [math.pr] 25 May 207 Abstract. The transition mechanism of
More information4. Ergodicity and mixing
4. Ergodicity and mixing 4. Introduction In the previous lecture we defined what is meant by an invariant measure. In this lecture, we define what is meant by an ergodic measure. The primary motivation
More informationNumerical Linear Algebra Homework Assignment - Week 2
Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.
More informationT.8. Perron-Frobenius theory of positive matrices From: H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003
T.8. Perron-Frobenius theory of positive matrices From: H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003 A vector x R n is called positive, symbolically x > 0,
More informationMarkov Chains. Andreas Klappenecker by Andreas Klappenecker. All rights reserved. Texas A&M University
Markov Chains Andreas Klappenecker Texas A&M University 208 by Andreas Klappenecker. All rights reserved. / 58 Stochastic Processes A stochastic process X tx ptq: t P T u is a collection of random variables.
More informationWaiting time distributions of simple and compound patterns in a sequence of r-th order Markov dependent multi-state trials
AISM (2006) 58: 291 310 DOI 10.1007/s10463-006-0038-8 James C. Fu W.Y. Wendy Lou Waiting time distributions of simple and compound patterns in a sequence of r-th order Markov dependent multi-state trials
More informationA New Look at Matrix Analytic Methods
Clemson University TigerPrints All Dissertations Dissertations 8-216 A New Look at Matrix Analytic Methods Jason Joyner Clemson University Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations
More informationThe Transition Probability Function P ij (t)
The Transition Probability Function P ij (t) Consider a continuous time Markov chain {X(t), t 0}. We are interested in the probability that in t time units the process will be in state j, given that it
More informationINTRODUCTION TO MARKOV CHAIN MONTE CARLO
INTRODUCTION TO MARKOV CHAIN MONTE CARLO 1. Introduction: MCMC In its simplest incarnation, the Monte Carlo method is nothing more than a computerbased exploitation of the Law of Large Numbers to estimate
More informationChapter 5. Continuous-Time Markov Chains. Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan
Chapter 5. Continuous-Time Markov Chains Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan Continuous-Time Markov Chains Consider a continuous-time stochastic process
More informationMath 203A - Solution Set 1
Math 203A - Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in
More informationGeometric Mapping Properties of Semipositive Matrices
Geometric Mapping Properties of Semipositive Matrices M. J. Tsatsomeros Mathematics Department Washington State University Pullman, WA 99164 (tsat@wsu.edu) July 14, 2015 Abstract Semipositive matrices
More informationSTA 294: Stochastic Processes & Bayesian Nonparametrics
MARKOV CHAINS AND CONVERGENCE CONCEPTS Markov chains are among the simplest stochastic processes, just one step beyond iid sequences of random variables. Traditionally they ve been used in modelling a
More information(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.
1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III
More informationLIMITING PROBABILITY TRANSITION MATRIX OF A CONDENSED FIBONACCI TREE
International Journal of Applied Mathematics Volume 31 No. 18, 41-49 ISSN: 1311-178 (printed version); ISSN: 1314-86 (on-line version) doi: http://dx.doi.org/1.173/ijam.v31i.6 LIMITING PROBABILITY TRANSITION
More informationThis operation is - associative A + (B + C) = (A + B) + C; - commutative A + B = B + A; - has a neutral element O + A = A, here O is the null matrix
1 Matrix Algebra Reading [SB] 81-85, pp 153-180 11 Matrix Operations 1 Addition a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn + b 11 b 12 b 1n b 21 b 22 b 2n b m1 b m2 b mn a 11 + b 11 a 12 + b 12 a 1n
More informationSome Definition and Example of Markov Chain
Some Definition and Example of Markov Chain Bowen Dai The Ohio State University April 5 th 2016 Introduction Definition and Notation Simple example of Markov Chain Aim Have some taste of Markov Chain and
More informationProperties of Matrices and Operations on Matrices
Properties of Matrices and Operations on Matrices A common data structure for statistical analysis is a rectangular array or matris. Rows represent individual observational units, or just observations,
More informationMarkov Processes Hamid R. Rabiee
Markov Processes Hamid R. Rabiee Overview Markov Property Markov Chains Definition Stationary Property Paths in Markov Chains Classification of States Steady States in MCs. 2 Markov Property A discrete
More informationHomework 4 due on Thursday, December 15 at 5 PM (hard deadline).
Large-Time Behavior for Continuous-Time Markov Chains Friday, December 02, 2011 10:58 AM Homework 4 due on Thursday, December 15 at 5 PM (hard deadline). How are formulas for large-time behavior of discrete-time
More informationT. Liggett Mathematics 171 Final Exam June 8, 2011
T. Liggett Mathematics 171 Final Exam June 8, 2011 1. The continuous time renewal chain X t has state space S = {0, 1, 2,...} and transition rates (i.e., Q matrix) given by q(n, n 1) = δ n and q(0, n)
More information25.1 Ergodicity and Metric Transitivity
Chapter 25 Ergodicity This lecture explains what it means for a process to be ergodic or metrically transitive, gives a few characterizes of these properties (especially for AMS processes), and deduces
More informationMS&E 321 Spring Stochastic Systems June 1, 2013 Prof. Peter W. Glynn Page 1 of 10
MS&E 321 Spring 12-13 Stochastic Systems June 1, 2013 Prof. Peter W. Glynn Page 1 of 10 Section 3: Regenerative Processes Contents 3.1 Regeneration: The Basic Idea............................... 1 3.2
More information1 Determinants. 1.1 Determinant
1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to
More information