Exercise 51 shows that E[β1X(0) = 0] = q(1 − p)/(q − p) − 1/p 1 6 Classification of States Depending on its transition probabilities, a Markov chain may visit
| Previous PDF | Next PDF |
[PDF] 9 Markov Chains: Introduction
Markov Chains: A discrete-time stochastic process X is said to be a Markov Chain if it has the Markov Property: Markov Property (version 1): For any s, i0, ,in−1 ∈ S and any n ≥ 1, P(Xn = sX0 = i0, ,Xn−1 = in−1) = P(Xn = sXn−1 = in−1)
[PDF] Markov Chains - Department of Statistics and Data Science
Exercise 51 shows that E[β1X(0) = 0] = q(1 − p)/(q − p) − 1/p 1 6 Classification of States Depending on its transition probabilities, a Markov chain may visit
[PDF] 5 Markov Chains
The process X is a Markov chain if it satisfies the Markov property: P(Xn+1 Show that every transition matrix on a finite state space has at least one closed
[PDF] Markov chains
(i) Show that the Xn form a Markov chain (ii) Find its transition probabilities Solution (i) Fix a time n ≥ 1 Suppose that you know that Xn = x The goal is to show
[PDF] Part IB - Markov Chains (Theorems with proof) - Dexter Chua
since P(X1 = · X0 = i) is a probability distribution function Theorem Let λ be a distribution (on S) and P a stochastic matrix The sequence X = (X0,X1,
[PDF] Introduction to Stochastic Processes - University of Kent
Theorem 2 25 Let X denote an irreducible, positive recurrent Markov chain Then X has a unique stationary distribution Proof: Existence has been shown in
[PDF] Chapter 8: Markov Chains
The transition diagram above shows a system with 7 possible states: state space Definition: The state space of a Markov chain, S, is the set of values that each
17 Markov Chains
Proof If the conditional distributions exist, then, by Theorem 17 9, the equation ( 17 4) is equivalent to X being a Markov chain Hence we only have to show that
[PDF] 01 Markov Chains
that X◦,X1,X2 ททท is a Markov chain with state space Z/n = {0,1,2,ททท ,n − 1} Show that the sequence of random variables Y◦,Y1,Y2,ททท where Yj = Sj
[PDF] show that [0
[PDF] show the mechanism of acid hydrolysis of ester
[PDF] show time zone cisco
[PDF] show ∞ n 2 1 n log np converges if and only if p > 1
[PDF] shredded workout plan pdf
[PDF] shredding diet
[PDF] shredding workout plan
[PDF] shrm furlough
[PDF] shuttle paris
[PDF] si clauses french examples
[PDF] si clauses french exercises
[PDF] si clauses french practice
[PDF] si clauses french practice pdf
[PDF] si present
Chapter 1
Markov Chains
A sequence of random variablesX
0 ,X 1 ,...with values in a countable setSis a Markov chain if at any timen, the future states (or values) Xn+1 ,X n+2 depend on the historyX 0 ,...,X n only through the present stateX n .Markovchains are fundamental stochastic processes that have many diverse applica-tions. This is because a Markov chain represents any dynamical system whose
states satisfy the recursionX n =f(X n1 ,Y n ),nτ1, whereY 1,Y 2 ...are independent and identically distributed (i.i.d.) andfis a deterministic func- tion. That is, the new stateX n is simply a function of the last state and an auxiliary random variable. Such system dynamics are typical of those for queuel engthsin call cen ters,s tresseso n materials, waiting times in produc- tion and service facilities, inventories in supply chains, parallel-processing software, water levels in dams, insurance funds, stock prices, etc. This chapter begins by describing the basic structure of a Markov chain and how its single-step transition probabilities determine its evolution. For in-stance, what is the probability of reaching a certain state, and how long doesit take to reach it? The next and main part of the chapter characterizes the
stationary or equilibrium distribution of Markov chains. These distributions are the basis of limiting averages of various cost and performance param- eters associated with Markov chains. Considerable discussion is devoted to branching phenomena, stochastic networks, and time-reversible chains. In-cluded are examples of Markov chains that represent queueing, productionsystems, inventory control, reliability, and Monte Carlo simulations.
Before getting into the main text, a reader would benefit by a brief review of conditional probabilities in Section 1.22 of this chapter and related material on random variables and distributions in Sections 1-4 in the Appendix. Therest of the Appendix, which provides more background on probability, wouldbe appropriate for later reading.
R. Serfozo,Basics of Applied Stochastic Processes,Probability and its Applications.
cSpringer-Verlag Berlin Heidelberg 2009 121MarkovChains
1.1 Introduction
This section introduces Markov chains and describes a few examples.A discrete-timestochastic process{X
n :n0}on a countable setS is a collection ofS-valued random variables deìned on a probability space (Ω,F,P). ThePis a probability measure on a family of eventsF(aσ-ìeld) in an event-spaceΩ. 1The setSis thestate spaceof the process, and the
valueX n Sis thestateof the process attimen.Thenmay represent a parameter other than time such as a length or a job number. Theìnite-dimensionaldistributions of the process are P{X 0 =i 0 ,...,X n =i n },i 0 ,...,i nS, n0.
These probabilities uniquely determine the probabilities of all events of the process. Consequently, two stochastic processes (deìned on dierent probabil- ity spaces or the same one) are equal in distribution if their ìnite-dimensional distributions are equal. Various types of stochastic processes are deìned by specifying the dependency among the variables that determine the ìnite- dimensional distributions, or by specifying the manner in which the process evolves over time (the system dynamics).A Markov chain is deìned as follows.
DeÞnition 1?A stochastic processX={X
n :n0}on a countable setS is aMarkov Chainif, for anyi,jSandn0, P{X n+1 =j|X 0 ,...,X n }=P{X n+1 =j|X n },(1.1) P{X n+1 =j|X n =i}=p ij .(1.2) Thep ij is the probability that the Markov chain jumps from stateito state j.Thesetransition probabilitiessatisfy jS p ij =1,iS,andthematrix P=(p ij )isthetransition matrixof the chain. Condition (1.1), called theMarkov property, says that, at any timen,the next stateX n+1 is conditionally independent of the pastX 0 ,...,X nŠ1 given the present stateX n . In other words, the next state is dependent on the past and present only through the present state. The Markov property is an elementary condition that is satisìed by the state of many stochastic phe- nomena. Consequently, Markov chains, and related continuous-time Markov processes, are natural models or building blocks for applications. Condition (1.2) simply says the transition probabilities do not depend on thetimeparametern; the Markov chain is therefore çtime-homogeneousé. If the transition probabilities were functions of time, the processX n would be a non-time-homogeneous Markov chain. Such chains are like time-homogeneous 1 Further details on probability spaces are in the Appendix. We follow the convention of not displaying the space (ffi,F,P) every time random variables or processes are introduced; it is mentioned only when needed for clarity.1.1 Introduction3
chains, but the time dependency introduces added accounting details that we will not address here. See Exercises 12 and 13 for further insights. Since the state spaceSis countable, we will sometimes label the states by integers, such asS={0,1,2,...}(orS={1,...,m}). Under this labeling, the transition matrix has the form P= p 00 p 01 p 02 p 10 p 11 p 12 p 20 p 21p 22
We end this section with a few preliminary examples. Example 2. Binomial Markov Chain.ABernoulli processis a sequence of independent trials in which each trial results in a success or failure with n denote the number of successes inntrials, forn1. By direct reasoning, it follows thatX n has a binomial distribution with parametersnandp: P{X n =k}= n k p k nŠk ,0kn.