Discrete-time stochastic processes

Since each individual generates descendants

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Unformatted text preview: st be calculated for all i before proceeding to calculate them for the next larger value of n. This also gives us fj j (n), although fj j (n) is not used in the iteration. 1 We say ‘essentially forms a renewal process’ because we haven’t yet specified the exact conditions upon which these returns to a given state form a renewal process. Note, however, that since we start the process in state j at time 0, the time at which the first renewal occurs is the same as the interval between successive renewals. 5.1. INTRODUCTION AND CLASSIFICATION OF STATES 199 Let Fij (n), for n ≥ 1, be the probability, given X0 = i, that state j occurs at some time between 1 and n inclusive. Thus, Fij (n) = n X fij (m). (5.3) m=1 For each i, j , Fij (n) is non-decreasing in n and (since it is a probability) is upper bounded by 1. Thus Fij (1), i.e., limn→1 Fij (n) must exist, and is the probability, given X0 = i, that state j will ever occur. If Fij (1) = 1, then, given X0 = i, it is certain (with probability 1) that the chain will eventually enter state j . In this case, we can define a random variable (rv) Tij , conditional on X0 = i, as the first passage time from i to j . Then fij (n) is the probability mass function of Tij and Fij (n) is the distribution function of Tij . If Fij (1) < 1, then Tij is a defective rv, since, with some non-zero probability, there is no first passage to j . Defective rv’s are not considered to be rv’s (in the theorems here or elsewhere), but they do have many of the properties of rv’s. The first passage time Tj j from a state j back to itself is of particular importance. It has the PMF fj j (n), the distribution function Fj j (n), and is a rv (as opposed to a defective rv) if Fj j (1) = 1, i.e., if the state eventually returns to state j with probability 1 given that it starts in state j . This leads to the definition of recurrence. Denition 5.2. A state j in a countable-state Markov chain is recurrent if Fj j (1) = 1. It is transient if Fj j (1) < 1. Thus each state j in a countable-state Markov chain is either recurrent or transient, and is recurrent if and only if (iff ) an eventual return to j occurs W.P.1, given that X0 = j . Equivalently, j is recurrent iff Tj j , the time to first return to j , is a rv. Note that for the special case of finite-state Markov chains, this definition is consistent with the one in Chapter 4. For a countably-infinite state space, however, the earlier definition is not adequate; for example, i and j communicate for all states i and j in Figure 5.1, but for p > 1/2, each state is transient (this is shown in Exercise 5.2, and further explained in Section 5.3). If state j is recurrent, and if the initial state is specified to be X0 = j , then Tj j is the integer time of the first recurrence of state j . At that recurrence, the Markov chain is in the same state j as it started in, and the discrete interval from Tj j to the next occurence of state j , say Tj j,2 has the same distribution as Tj j and is clearly independent of Tj j . Similarly, the sequence of successive recurrence intervals, Tj j , Tj j,2 , Tj j,3 , . . . is a sequence of IID rv’s. This sequence of recurrence intervals2 is then the sequence of inter-renewal intervals of a renewal process, where each renewal interval has the distribution of Tj j . These inter-renewal intervals have the PMF fj j (n) and the distribution function Fj j (n). Since results about Markov chains depend very heavily on whether states are recurrent or transient, we will look carefully at the probabilities Fij (n). Substituting (5.2) into (5.3), we 2 Note that in Chapter 3 the inter-renewal intervals were denoted X1 , X2 , . . . , whereas here X0 , X1 , . . . , is the sequence of states in the Markov chain and Tj j , Tj j,2 , . . . , is the sequence of inter-renewal intervals. 200 CHAPTER 5. COUNTABLE-STATE MARKOV CHAINS obtain Fij (n) = Pij + X k6=j Pik Fkj (n − 1); n > 1; Fij (1) = Pij . (5.4) P In the expression Pij + k6=j Pik FkjP − 1), note that Pij is the probability that state j is (n entered on the first transition, and k6=j Pik Fkj (n − 1) is the sum, over every other state k, of the joint probability that k is entered on the first transition and that j is entered on one of the subsequent n − 1 transitions. For each i, Fij (n) is non-decreasing in n and upper bounded by 1 (this can be seen from (5.3), and can also be established directly from (5.4) by induction). Thus it can be shown that the limit as n → 1 exists and satisfies Fij (1) = Pij + X k6=j Pik Fkj (1). (5.5) There is not always a unique solution to (5.5). That is, the set of equations xij = Pij + X Pik xkj ; k6=j all i ≥ 0 (5.6) always has a solution in which xij = 1 for all i ≥ 0, but if state j is transient, there is another solution in which xij is the true value of Fij (1) and Fj j (1) < 1. Exercise 5.1 shows that if (5.6) is satisfied by a set of non-negative numbers {xij ; 1 ≤ i ≤ J }, then...
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