Discrete-time stochastic processes

# z1 zi 1 i n 775 1 i n 774 equations

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Unformatted text preview: exp n lim ∞ (r) − r∈(r− ,r+ )→r+ if α/n = ∞ 0 (ro ) for ro &lt; r+ rα n ∏æ otherwise. (7.23) 7.5. THRESHOLDS, STOPPING RULES, AND WALD’S IDENTITY 0 ∞ (r) − rα/n r r∗ = r+ ∞ (r) 291 ∞ (r∗ ) r∗ − ∞ (r∗ )(n/α) slope = α/n ∞ (r∗ ) − r∗ α/n Figure 7.5: Graphical minimization of ∞ (r) − (α/n)r for the case where r+ &lt; 1. As before, for any r &lt; r+ , ∞ (r) − rα/n is found by drawing a line of slope (α/n) from the point (r, ∞ (r)) to the vertical axis. The minimum occurs when the line of slope α/n is tangent to the curve or when it touches the curve at r = r∗ . If we extend the deﬁnition of r∗ as the supremum of r such that ∞ (r) ≤ 0, then Pr {Sn ≥ α} ≥ exp(−r∗ α) still holds for arbitrary α &gt; 0, n ≥ 1. The next section establishes Wald’s identity, which shows, among other things, that if X &lt; 0, then exp(−r∗ α) is an upper bound (and a reasonable approximation) to the probability that the walk ever crosses a threshold at α &gt; 0. Note that we have already found an upper S bound to Pr {Sn ≥ α} for any α &gt; 0, n ≥ 1, but this new result bounds Pr { n {Sn ≥ α}} for any α &gt; 0. Both the threshold-crossing bounds in this section and Wald’s identity in the next suggest that for large n or large α, the most important parameter of the IID rv’s X making up the walk is the positive root r∗ of ∞ (r), rather than the mean, variance, or other moments of X . As a prelude to developing these large deviation results about threshold crossings, we deﬁne stopping rules in a way that is both simpler and more general than the treatment in Chapter 3. 7.5 Thresholds, stopping rules, and Wald’s identity The following lemma shows that a random walk with two thresholds, say α &gt; 0 and β &lt; 0, eventually crosses one of the threshold. Figure 7.6 illustrates two sample paths and how they cross thresholds. More speciﬁcally, the random walk ﬁrst crosses a thresholds at trial n if β &lt; Si &lt; α for 1 ≤ i &lt; n and either Sn ≥ α or Sn ≤ β . The lemma shows that this random number of trials N is ﬁnite with probability 1 (i.e., N is a rv) and that N has moments of all orders. Lemma 7.1. Let {Xi ; i ≥ 1} be IID and not identical ly 0. For each n ≥ 1, let Sn = X1 + · · · + Xn . Let α &gt; 0 and β &lt; 0 be arbitrary real numbers , and let N be the smal lest n for which either Sn ≥ α or Sn ≤ β . Consider a random walk with two thresholds, α &gt; 0 and β &lt; 0, and assume that X is not identical ly zero. Then N is a random variable (i.e., limm→1 Pr {N ≥ m} = 0) and has ﬁnite moments of al l orders. 292 CHAPTER 7. α β S1 r S2 r RANDOM WALKS, LARGE DEVIATIONS, AND MARTINGALES S3 r S4 r S5 r S6 r α β S1 r S2 r S3 r S4 r S5 r S6 r Figure 7.6: Two sample paths of a random walk with two thresholds. In the ﬁrst, the threshold at α is crossed at N = 5. In the second, the threshold at β is crossed at N =4 Proof: Since X is not identically 0, there is some n for which either Pr {Sn ≤ −α + β } &gt; 0 or for which Pr {Sn ≥ α − β } &gt; 0. For any such n, let ε = max[Pr {Sn ≤ −α + β } , Pr {Sn ≥ α − β }]. For any integer k ≥ 1, given that N &gt; n(k − 1), and given any value of Sn(k−1) in (β , α), a threshold will be crossed by time nk with probability at least ε. Thus, Pr {N &gt; nk | N &gt; n(k − 1)} ≤ 1 − ε, Iterating on k, Pr {N &gt; nk} ≤ (1 − ε)k . This shows that N is ﬁnite with probability 1 and that Pr {N ≥ j } goes to 0 at least geometrically in j . It follows that the moment generating function gN (r) of N is ﬁnite in a region around r = 0, and that N has moments of all orders. 7.5.1 Stopping rules In this section, we start with a deﬁnition of stopping rules that is more fundamental and quite diﬀerent from that in Chapter 3. We then use this deﬁnition to establish Wald’s identity, which is the basis for all of our subsequent results about random walks and threshold crossings. First consider a simple example. Consider a sequence {Xn ; n ≥ 1} of binary random variables taking on only the values ±1. Suppose we are interested in the ﬁrst occurrence of the string (+1, −1), and we view this condition as a stopping rule. Figure 7.7 illustrates this stopped process by viewing it as the truncation of a tree of possible sequences. Aside from the complexity of the tree, the same approach can be taken when considering a random walk with a stopping rule that stops at the ﬁrst trial in which the random walk reaches either α &gt; 0 or β &lt; 0. In this case also, the stopping node is the initial segment for which the ﬁrst crossing occurs at the ﬁnal trial of that segment. 7.5. THRESHOLDS, STOPPING RULES, AND WALD’S IDENTITY r 1 -1 r r 1 -1 1 -1 r s r r r s r s r r 293 r s r s r s r ✘ r Figure 7.7: A tree representing the collection of binary (1, -1) sequences, with a stopping rule viewed as a pruning of the tree. The particular stopping rule here is to stop on the ﬁrst occurrence of the string (+1, −1). The leaves of the tree (i.e.,...
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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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