Discrete-time stochastic processes

This result namely that past and future are

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Unformatted text preview: h Pj m > 0 and Pmi > 0. For a contradiction, again assume that Fij (1) < 1. From (5.5), X X Fmj (1) = Pmj + Pmk Fkj (1) < Pmj + Pmk = 1, k6=j k6=j where the strict inequality follows since Pmi Fij (1) < Pmi . This is a contradiction, since m is accessible from j in one step, and thus Fmj (1) = 1. It follows that every i accessible from j in two steps satisfies Fij (1) = 1. Extending the same argument for successively larger numbers of steps, the conclusion of the lemma follows. Lemma 5.4. Let {Nij (t); t ≥ 0} be the counting process for transitions into state j up to time t for a Markov chain given X0 = i 6= j . Then if i and j are in the same recurrent class, {Nij (t); t ≥ 0} is a delayed renewal process. Proof: From Lemma 5.3, Tij , the time until the first transition into j , is a rv. Also Tj j is a rv by definition of recurrence, and subsequent intervals between occurrences of state j are IID, completing the proof. If Fij (1) = 1, we have seen that the first-passage time from i to j is a rv, i.e., is finite with probability 1. In this case, the mean time T ij to first enter state j starting from state i is of interest. Since Tij is a non-negative random variable, its expectation is the integral of its complementary distribution function, T ij = 1 + 1 X (1 − Fij (n)). (5.7) n=1 It is possible to have Fij (1) = 1 but T ij = 1. As will be shown in Section 5.3, the chain in Figure 5.1 satisfies Fij (1) = 1 and T ij < 1 for p < 1/2 and Fij (1) = 1 and T ij = 1 for p = 1/2. As discussed before, Fij (1) < 1 for p > 1/2. This leads us to the following definition. 5.1. INTRODUCTION AND CLASSIFICATION OF STATES 203 Definition 5.3. A state j in a countable-state Markov chain is positive-recurrent if Fj j (1) = 1 and T j j < 1. It is null-recurrent if Fj j (1) = 1 and T j j = 1. Each state of a Markov chain is thus classified as one of the following three types — positiverecurrent, null-recurrent, or transient. For the example of Figure 5.1, null-recurrence lies on a boundary between positive-recurrence and transience, and this is often a good way to look at null-recurrence. Part f ) of Exercise 6.1 illustrates another type of situation in which null-recurrence can occur. Assume that state j is recurrent and consider the renewal process {Nj j (t); t ≥ 0}. The limiting theorems for renewal processes can be applied directly. From the strong law for renewal processes, Theorem 3.1, lim Nj j (t)/t = 1/T j j t→1 with probability 1. (5.8) From the elementary renewal theorem, Theorem 3.4, lim E [Nj j (t)/t] = 1/T j j . t→1 (5.9) Equations (5.8) and (5.9) are valid whether j is positive-recurrent or null-recurrent. Next we apply Blackwell’s theorem to {Nj j (t); t ≥ 0}. Recall that the period of a given state j in a Markov chain (whether the chain has a countable or finite number of states) is the greatest common divisor of the set of integers n > 0 such that Pjn > 0. If this period j is d, then {Nj j (t); t ≥ 0} is arithmetic with span d (i.e., renewals occur only at times that are multiples of d). From Blackwell’s theorem in the arithmetic form of (3.20), lim Pr {Xnd = j | X0 = j } = d/T j j . n→1 (5.10) If state j is aperiodic (i.e., d = 1), this says that limn→1 Pr {Xn = j | X0 = j } = 1/T j j . Equations (5.8) and (5.9) suggest that 1/T j j has some of the properties associated with a steady-state probability of state j , and (5.10) strengthens this if j is aperiodic. For a Markov chain consisting of a single class of states, all positive-recurrent, we will strengthen this association further in Theorem 5.4 by showing that there is a unique P ady-state ste distribution, {P, j ≥ 0} such that πj = 1/T j j for all j and such that πj = i πi Pij for πj all j ≥ 0 and j πj = 1. The following theorem starts this development by showing that (5.8-5.10) are independent of the starting state. Theorem 5.2. Let j be a recurrent state in a Markov chain and let i be any state in the same class as j . Given X0 = i, let Nij (t) be the number of transitions into state j by time t and let T j j be the expected recurrence time of state j (either finite or infinite). Then lim Nij (t)/t = 1/T j j t→1 with probability 1 lim E [Nij (t)/t] = 1/T j j . t→1 (5.11) (5.12) If j is also aperiodic, then lim Pr {Xn = j | X0 = i} = 1/T j j . n→1 (5.13) 204 CHAPTER 5. COUNTABLE-STATE MARKOV CHAINS Proof: Since i and j are recurrent and in the same class, Lemma 5.4 asserts that {Nij (t); t ≥ 0} is a delayed renewal process for j 6= i. Thus (5.11) and (5.12) follow from Theorems 3.9 and 3.10 of Chapter 3. If j is aperiodic, then {Nij (t); t ≥ 0} is a delayed renewal process for which the inter-renewal intervals Tj j have span 1 and Tij has an integer span. Thus, (5.13) follows from Blackwell’s theorem for delayed renewal processes, Theorem 3.11. For i = j , Equations (5.11-5.13) follow from (5.8-5.10), completing the proof. Theorem 5.3. Al l states in the same class of a Markov chain are of the same type — either al l positive-recurrent, al l nul l-recurrent, o...
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