Otherwise it is reducible. What is the period? 2. By the previous proposition, we know that also j → i. If it is a finite-state chain, it necessarily has to be recurrent. • All states of a finite irreducible Markov chain are recurrent. A state has period \(k\) if it must return to that state in multiples of \(k\) moves. Let's consider a finite Markov chain. At each time step the population grows by one, lowers by one, or stays the same.
We have the following transition function: In this lecture we shall brie y overview the basic theoretical foundation of DTMC. the Markov chain in question is either 1) not finite, 2) not irreducible, and 3) not aperiodic. An example of a recurrent Markov chain is the symmetric random walk on the integer lattice on the line or plane. Chain Monte Carlo example. The study of discrete-time Markov chains, particularly the limiting behavior, depends critically on the random times between visits to a given state. It is known that a Markov chain is irreducible if and only if any two states intercommunicate. An irreducible Markov chain Xn on a finite state space n!1 n = g=ˇ( T T If the state space is finite and all states communicate (that is, the Markov chain is irreducible) then in the long run, regardless of the initial condition, the Markov chain must settle into a steady state. Example: epidemics. As we will see shortly, irreducibility is a desirable property in the sense that it can simplify analysis of the limiting behavior. We now show that null recurrent chains, like transient ones, do not have stationary distributions. In light of this proposition, we can classify each class, and an irreducible Markov chain, as recurrent or transient. Continuous-Time Markov Chains 17.1. An irreducible Markov chain Xn on a finite state space n!1 n = g=ˇ( T T In an irreducible Markov Chain, the process can go from any state to any state, whatever be the number of steps it requires. For definiteness assume X = 1. Classical example would be a random walk with a bias: The random walker drifts off to infinity, eventually doesn't return. An example of a stationary irreducible transient Markov chain {X n n <∞} with {itX n n < <∞ ergodic but {X n n ≦0 nonergodic is given. Recurrence is a class property: all states in a class are either recurrent or transient. 11.2.6 Stationary and Limiting Distributions. That this can be the case has already been implicitly . Examples: 1. Irreducible Markov chains. It does this by constructing an appropriate transition probability for ˇ An irreducible Markov Chain is a Markov Chain with with a path between any pair of states. For the above example, the Markov chain resulting from the first transition matrix will be irreducible while the chain resulting from the second matrix will be reducible into two clusters: one including states x 1 and x (a) 1 0 0 0 1 0 1/3 1/3 1/3 (b) 0 1/2 1/2 1 0 0 1 0 0 (c) 0 1 0 0 0 1 1/2 1/2 0 (d) 1 0 0 In particular, if the chain is irreducible, then either all states are recurrent or all are transient. Transient and recurrent states. Abstract. (Example 7.1) that the state space of this Markov chain can be decomposed into the set of transient states \(T=\{3,4\}\) and two . If we now consider the rat in the closed maze, S = {1,2,3,4}, then we see that there is only one communication class C = {1,2,3,4} = S: all states communicate. The n n →∞ = p limiting state probabilities π p limit exists and is independent of the initial state-probability vector, 2. A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. A . Theorem C. An irreducible null recurrent Markov chain has no stationary distribution. one communication class C = {1,2,3,4}= S: all states communicate. FSDT Markov chains that aren't irreducible but do have a single closed communication class. Draw (or sketch) the transition graphs and examine whether the chains are irreducible. In this distribution, every state has positive probability. A continuous-time process is called a continuous-time Markov chain (CTMC). Finite Markov Chain [ ] [ ] For a finite Markov chain with an initial state-probability vector 0 the , if they exist, are the elements of the vector lim . • Corollary 4.2: If state i is recurrent and state i com-municates with state j, then state j is recurrent. A continuous-time process is called a continuous-time Markov chain (CTMC). This is an example of what is called an irreducible Markov chain. The ergodic theorem for a Markov chain with a denumerable . A Markov chain is said to be irreducible if it has only one communicating class. )all states in an irreducible Markov chain are . Formally, Theorem 3. A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). . The Markov chain mc is irreducible if every state is reachable from every other state in at most n - 1 steps, where n is the number of states (mc.NumStates).This result is equivalent to Q = (I + Z) n - 1 containing all positive elements. Answer: People are usually more interested in cases when Markov Chain's do have a stationary distribution. Proof. 1. The central problem discussed in this paper is that of deciding the recurrence or transience of 'random walk' on a regular grid in the plane, for example, the equilateral triangular grid, or the hexagonal grid. ∞, and transient if P∞ n=0 P n i,i < ∞. Irreducible Markov chains I A MC is called irreducible if it has only one class I All states communicate with each other I If MC also has finite number of states the single class is recurrent I If MC infinite, class might be transient I When it has multiple classes (not irreducible) I Classes of transient states T 1, T 2, . Let us rst look at a few examples which can be naturally modelled by a DTMC. A simple random walk on Z is a Markov chain with state space E= Z and Moreover, ← − both chains X and Xe can start at state 0. In that case, we can talk of the chain itself being transient or recurrent. A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). Lemma 2.7.11. The "Gambler's Ruin" Markov chain is periodic, because, for example, you can only ever return to state 0 at even time-steps: gcdftjPr[X t= 0jX 0 = 0] >0g= 2: Fact 6. is determined by solving and ! Classify the states. All the states of an irreducible chain, whether finite or denumerable, are of the same type: all transient, all null persistent, or all nonnull persistent. Birth-Death Example 1-p 1-p p p 1-p p 0 1 i p! Consider the following transition matrices. Ergodic Markov chain. Identify the transient and recurrent states, and the irreducible closed sets in the Markov chains. Figure 1.2 shows a Markov Chain with three communicating classes: f1g, f2;3;4gand f5g.
There are a number of ways to see this is transient; one is to note that it can be realized as X n = X 0 + ξ 1 + ⋯ + ξ n where the ξ i are iid biased coin flips . 4. The rat in the open maze yields a Markov chain that is not irreducible; there are two communication classes C 1 = f1;2;3;4g;C 2 = f0g. The chain is non-null i P i x i < 1. Proof. Decompose a branching process, a simple random walk, and a random walk on a nite, disconnected graph. DEF 23.20 (Markovian coupling) A Markovian coupling of a transition proba-bility pis a Markov chain f(X n;Y Not all states can be transient. Any irreducible Markov chain has a unique stationary distribution. • Corollary 4.3: A finite state Markov chain cannot have all transient states. The steady state vector !
For any non-irreducible Markov chain, we can identify the recurrent classes using the following process I Create directed edges between any two nodes that have a non-zero transition probability between them. . 3. A standard example is asymmetric random walk on the integers: consider a Markov chain with state space Z and transition probability p ( x, x + 1) = 3 / 4, p ( x, x − 1) = 1 / 4. π ( n) = [ P ( X n = 0) P ( X n = 1) ⋯] as n . State 5 is an absorbing state since it is impossible to leave it. Putting everything so far together, we have the following classification: non-closed classes are transient; If all the states in the Markov Chain belong to one closed communicating class, then the chain is called an irreducible Markov chain. I is the n-by-n identity matrix. pq = > 0 The MC is irreducible and null recurrent. 2. - For any irreducible and finite-state Markov chain, all states are recurrent. 1 Contents 1 Introduction 2 2 Markov Chains 2 . Proposition. If state i is currerent and i ↔ j, then j is curcerent. The rat in the closed maze yields a recurrent Markov chain. The die is biased and side j of die number i appears with probability P ij.
A state iis called recurrent if it is not transient. There is some possibility (a nonzero probability) that a process beginning in a transient state will never return to that state. A criterion for transience We proved this from first principles in Question 3 on Problem Sheet 3 . Clearly if the state space is nite for a given Markov chain, then not all the states can be We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains.
Decompose a branching process, a simple random walk, and a random walk on a nite, disconnected graph. The following is an example of an ergodic Markov Chain 13 In that case, we can talk of the chain itself being transient or recurrent. View Continuous-Time Markov Chains (3).pdf from PSTAT 160b at University of California, Santa Barbara. probability of never returning to i. written as a Markov chain whose state is a vector of k consecutive words. De nition: an irreducible closed set C is a closed set such that x!yfor all choices x;y2C. Formally, Theorem 3. It is shown that transient graphs for the simple random walk do not admit a nearest neighbor transient Markov chain (not necessarily a reversible one), that crosses all edges with positive probability, while there is such chain for the square grid $\mathbb{Z}^2$. The nature of these random times leads to a fundamental dichotomy of the states. A Markov chain is said to be Irreducible if we can go from one state to another in a single or more than one step. Markov Chains - 3 Some Observations About the Limi • The behavior of this important limit depends on properties of states i and j and the Markov chain as a whole. Example 5. Theorem: In an irreducible closed set, either all states are transient or all states are recurrent. Invariant distributions Suppose we observe a finite-state Markov chain over a long period of time. The transition matrix for four-state Markov chain is P=0 1/2 0 1/2 1/4 0 3/4 0 0 3/4 0 1/4 1/2 0 1/2 0 Show that the chain is irreducible, positive recurrent and periodic. If all states are aperiodic, then the Markov chain is aperiodic. In a recurrent Markov chain there are no inessential states and the essential states decompose into recurrent classes. Suppose each infected individual has some chance of contacting each susceptible individual in each time interval, before becoming removed (recovered or hospitalized). A Markov chain with one transient state and two recurrent states A stochastic process contains states that may be either transient or recurrent; transience and recurrence describe the likelihood of a process beginning in some state of returning to that particular state. Markov Chains - 10 Irreducibility • A Markov chain is irreducible if all states belong to one class (all states communicate with each other). In that case, we can talk of the chain itself being transient or recurrent. If \(k=1\), the state is aperiodic. Example 1.1 (Gambler Ruin Problem).
That this can be the case has already . 1. Proposition. An irreducible Markov chain is one where x!yfor all x;y2. An irreducible chain has all recurrent or transient states. Thus from the above results we note that all the states of an irreducible chain are either transient or recurrent, as desired. Discrete Time Markov Chains 1 Examples Discrete Time Markov Chain (DTMC) is an extremely pervasive probability model [1]. But for infinite state Markov chains, it is possible to have closed transient classes and even irreducible Markov chains that are transient. In your question if A is a set of transient states, then it is possible to escape to a state outside A (unless the Markov chain has infinite states) and thus for a finite Markov chain such a set of states would not be called irreducible. ⇒ an irreducible MC has only one class, which is necessarily closed. A simple example for a non-irreducible Markov chain can be given by our well-known model for the weather forecast where and If or , then the corresponding Markov chain is clearly not irreducible and therefore by Theorem 2.9 not ergodic. Recurrent and Transient States A state isrecurrentif the process will return to the state again with probability = 1. Transience and Recurrence for Discrete-Time Chains. If a Markov chain is irreducible then we also say that this chain is "ergodic" as it verifies the following ergodic theorem. If the chain is irreducible and persistent, there exists a un ique (up to a multiplicative constant) positive root x of x = xP. value of the MC as in (7), it follows that state 0 is always re-visited making it positive recurrent. either all states are recurrent or all are transient. Now according to Wikipedia: A state i is said to be transient if, given that we start in state i, there is a non-zero probability that we will never return to i. - If i and j are recurrent and belong to different classes, then p(n) ij=0 for all n. - If j is transient, then for all i.Intuitively, the More specifically, we would like to study the distributions. Recurrence and transience of random walks Example 2.8.1. Here, we would like to discuss long-term behavior of Markov chains. Any irreducible chain on a nite state space is non-null. Invariant distributions Suppose we observe a nite-state Markov chain over a long period of time. Part two of a problem concerning transience, recurrence, and closed irreducible sets of markov chains. Limit distribution of ergodic Markov chains Theorem For an ergodic (i.e., irreducible, aperiodic and positive recurrent) MC, lim n!1P n ij exists and is independent of the initial state i, i.e., ˇ j = lim n!1 Pn ij Furthermore, steady-state probabilities ˇ They key result is that A Markov Chain has a stationary distribution if and only if at least one state is . Such a process is a Markov chain, but The chain is null persistent. Give reasons f. A state istransientotherwise. Markov chain Monte Carlo 20/75 I Now suppose we are interested in sampling from a distribution ˇ(e.g., the unnormalized posterior) I Markov chain Monte Carlo (MCMC) is a method that samples from a Markov chain whose stationary distribution is the target distribution ˇ. The ideas of stationary distributions can also be extended simply to Markov chains that are reducible (not irreducible; some states don't communicate) if the Markov chain can be expressed as a union of closed communication classes (i.e., the letters If it is a nite-state chain, it necessarily has to be recurrent. Not all of our theorems will be if and only if's, but they are still illustrative. If it is a finite-state chain, it necessarily has to be recurrent. Corollary: In an irreducible closed nite set, all states are recurrent. De nition 8. The paper by Schweitzer you have cited deals exclusively with finite Markov chains. 13 Although the chain does spend 1/3 of the time at each state, the transition MCs with more than one class, may consist of both closed and non-closed classes: for the previous example 2.4.1 Birth and Death Chain Both queueing systems and populations are modeled by a birth and death chain. Exercise. The period of a state iin a Markov chain is the greatest common divisor of the possible numbers of steps it can take to return to iwhen starting at i. If the state space is finite and all states communicate (that is, the Markov chain is irreducible) then in the long run, regardless of the initial condition, the Markov chain must settle into a steady state. The zero-pattern matrix of the transition matrix P (mc.P) is Z ij = I(P ij > 0), for all i,j. Moreover P2 = 0 0 1 1 0 0 0 1 0 , P3 = I, P4 = P, etc. A coupling of Markov chains with transition probability pis a Markov chain f(X n;Y n)gon S Ssuch that both fX ngand fY ngare Markov chains with transition probability p. For our purposes, the following special type of coupling will suffice. A state in a Markov chain is said to be Periodic if returning to it requires a multiple of some integer larger than 1, the greatest common divisor of all the possible return path lengths will be the period of that state. (i) For a monotone irreducible stochastic kernel P , the Λ−intertwining Markov chain Xe has N as absorbing state and it is a sharp dual of ← − X . De nition A Markov chain is called irreducible if and only if all states belong to one communication class. ∞, and transient if P∞ n=0 P n i,i < ∞. . Finally, the Markov chain is said to be irreducible it it consists of a single communicating class. Transitivity follows by composing paths. In particular, we would like to know the fraction of times that the Markov chain spends in each state as n becomes large. exists such that $ j > 0 and where M j is the mean recurrence time of state j! (Infinite State Spaces) There is an analog of the theorem that applies to Markov chains with an infinite state space, which you will see on Problem Set 7. Assume that we have an application f(.) Lemma 2.2 actsF about currcerence. A Markov chain is irreducible if all the states communicate. • If a Markov chain is not irreducible, it is called reducible. • Corollary 4.3: A finite state Markov chain cannot have all transient states. Invariant distributions Suppose we observe a finite-state Markov chain over a long period of time. If we are interested in investigating questions about the Markov chain in L ≤ ∞ units of time (i.e., the subscript l ≤ L), then we are looking at all possible sequences 1k . A Markov chain is called reducible if Answer: 1P0 ii = P(X0 = i|X0 = i) = 1, a trivial fact. • If state i is recurrent and state i does not communicate with state j, then Pij 0 - when a process enters a recurrent class of states it can never leave that class. Consider an irreducible, recurrent Markov chain with an arbitrary initial distribution . This is an example of what is called an irreducible Markov chain.
Properties of states and Markov chains¶ A Markov chain is irreducible if it is possible to get from any state to any state.
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