Discrete-time stochastic processes

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Unformatted text preview: in t, and thus has either a finite or infinite limit as t → 1. For each n, Pr {N (t) ≥ n} ≥ 1/2 for large enough t, and therefore E [N (t)] ≥ n/2 for all such t. Thus E [N (t)] can have no finite limit, and limt→1 E [N (t)] = 1. For any given t > 0, the random variable N (t) is the number of renewal epochs in the interval (0, t]. The random variable SN (t) is then the epoch at which renewal N (t) occurs, i.e., the latest renewal epoch before or equal to time t. Similarly SN (t)+1 is the first arrival epoch after time t (see Figure 3.2). Thus we have the inequalities SN (t) SN (t)+1 t ≤ < . N (t) N (t) N (t) (3.1) From lemma 3.1, limt→1 N (t) = 1 with probability 1. Assuming that X < 1, the strong law of large numbers (Theorem 1.5) asserts that limn→1 Sn /n = X with probability 1. For any sample function (i.e., sample point ω ), SN (t) (ω )/N (t, ω ) runs through the same sequence of values with increasing t as Sn (ω )/n runs through with increasing n. Thus letting ≠0 be the set of sample points of ω for which both limn→1 Sn /n = X and limt→1 N (t) = 1, we 3.2. STRONG LAW OF LARGE NUMBERS FOR RENEWAL PROCESSES 95 have limt→1 SN (t) /N (t) = X for all sample points in ≠0 . In the same way lim t→1 SN (t)+1 SN (t)+1 N (t) + 1 = lim =X t→1 N (t) + 1 N (t) N (t) Slope = Slope = N (t) t for all ω ∈ ≠0 . (3.2) N (t) SN (t) ✘✘✘ ❅ ❄ ✘✘✘ SN (t)+1 ❅ ✘✘✘ ❘ ❅ ❆ ✘✘✘ ❆ ✘✘✘ ❆ ✘✘ t ✘✘✘ N (t) 0 SN (t)+1 Slope = N (t) S1 SN (t) Figure 3.2: Comparison of N (t)/t with N (t) SN (t) and N (t) SN (t)+1 . Since t/N (t) in (3.1) lies between two random variables both converging to X for all sample points in ≠0 , we see that limt→1 t/N (t) = X for all sample functions in ≠0 , i.e., with probability 1. Since X must be greater than 0, it follows that limt→1 N (t)/t = 1/X for all sample points in ≠0 . This proves the following strong law for renewal processes. Theorem 3.1 (Strong Law for Renewal Processes). For a renewal process with mean inter-renewal interval X , limt→1 N (t)/t = 1/X with probability 1. This theorem is also true if the mean inter-renewal interval is infinite; this can be seen by a truncation argument (see Exercise 3.3). We could also prove a™weak law for N (t) (i.e., we © could show that for any ≤ > 0, limt→1 Pr |N (t)/t − 1/X | ≥ ≤ = 0). This could be done by using the weak law of large numbers for Sn (Theorem 1.3) and the fact that the event Sn ≤ t is the same as N (t) ≥ n. Such a derivation is tedious, however, and illustrates that the strong law of large numbers is often much easier to work with than the weak law. We shall not derive the weak law here, since the strong law for renewal processes implies the weak law and it is the strong law that is most often useful. Figure 3.3 helps give some appreciation of what the strong law for N (t) says and doesn’t say. The strong law deals with time-averages, limt→1 N (t, ω )/t, for individual sample points ω ; these are indicated in the figure as horizontal averages, one for each ω . It is also of interest to look at time and ensemble-averages, E [N (t)/t], shown in the figure as vertical averages. N (t, ω )/t is the time-average number of renewals from 0 to t, and E [N (t)/t] averages also over the ensemble. Finally, to focus on arrivals in the vicinity of a particular time t, it is of interest to look at the ensemble-average E [N (t + δ ) − N (t)] /δ . Given the strong law for N (t), one would hypothesize that E [N (t)/t] approaches 1/X as t → 1. One might also hypothesize that limt→1 E [N (t + δ ) − N (t)] /δ = 1/X , sub ject to some minor restrictions on δ . These hypotheses are correct and are discussed in detail in what follows. This equality of time-averages and limiting ensemble-averages for renewal processes carries over to a large number of stochastic processes, and forms the basis of ergodic theory. These results are important for both theoretical and practical purposes. It is sometimes easy to find time averages (just like it was easy to find the time-average 96 CHAPTER 3. RENEWAL PROCESSES N (t, ω )/t from the strong law of large numbers), and it is sometimes easy to find limiting ensemble-averages. Being able to equate the two then allows us to alternate at will between time and ensemble-averages. Ensemble Average at t Time and ensemble Average over (0, τ ) (1/δ )E [N (t + δ ) − N (t)] N (t,ω3 ) t ✲ Time Ave. at ω1 0 N (t,ω2 ) t ✲ Time Ave. at ω2 0 N (t,ω1 ) t 0 ✲ Time Ave. at ω3 τ t Figure 3.3: Time average at a sample point, time and ensemble average from 0 to a given τ , and the ensemble-average in an interval (t, t + δ ]. Note that in order to equate time-averages and limiting ensemble-averages, quite a few conditions are required. First, the time average must exist in the limit t → 1 with probability one and have a fixed value with probability one; second, the ensemble-average...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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