Discrete-time stochastic processes

Given a possibly defective stopping time n for a

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Unformatted text preview: the nodes at which stopping occurs) are marked with large dots and the intermediate nodes (the other nodes) with small dots. Note that each leaf in the tree has a oneto-one correspondence with an initial segment of the tree, so the stopping nodes can be unambiguously viewed either as leaves of the tree or initial segments of the sample sequences. Note that in both of these examples, the stopping rule determines which initial segment of any given sample sequence satisifes the rule. The distribution of each Xn , and even whether or not the sequence is IID, is usually not relevant for defining these stopping rules. In other words, the conditions about statistical independence used in Chapter 3 for the indicator functions of stopping rules is quite unnatural for most applications. The essence of a stopping rule, however, is illustrated quite well in Figure 7.7. If one stops at some initial segment of a sample sequence, then one cannot stop again at some longer initial segment of the same sample sequence. This leads us to the following definitions of stopping nodes, stopping rules, and stopping times. Definition 7.2 (Stopping nodes). Given a sequence {Xn ; n ≥ 1} of rv’s, a col lection of stopping nodes is a col lection of initial segments of the sample sequences of {Xn ; n ≥ 1}. If an initial segment of one sequence is a stopping node, then it is a stopping node for al l sequences with that same initial segment. Also, no stopping node can be an initial segment of any other stopping node. This definition is less abstract when each Xn is dicrete with a finite number, say m of possible values. In this case, as illustrated in Figure 7.7, the set of seqeunces is represented by a tree in which each node has one branch coming in from the root and m branches going out. Each stopping node corresponds to ‘pruning’ the tree at that node. All the sequences with that given initial segment can then be ignored since they all have that same initial 294 CHAPTER 7. RANDOM WALKS, LARGE DEVIATIONS, AND MARTINGALES segment, i.e., stopping node. In this sense, every ‘pruning’ of the tree corresponds to a collection of stopping nodes. In information theory, such a collection of stopping nodes is called a prefix-free source code. Each segment corresponding to a stopping node is used as a codeword for some given message. If a sequence of consecutive segments is transmitted, a receiver can parse the incoming letters into segments by using the fact that no stopping node is an initial segment of any other stopping node. Definition 7.3 (Stopping rule and stopping time). A stopping rule for {Xn ; n ≥ 1} is a rule that determines a col lection of stopping nodes. A stopping time is a perhaps defective rv whose value, for a sample sequence with a stopping node, is the length of the initial segment for that node. Its value, for a sample sequence with no stopping node, is infinite. For most interesting stopping rules, sample sequences exist that have no stopping nodes. For the example of a random walk with two thresholds, there are many sequences that stay inside the thresholds forever. As shown by Lemma 7.1 however, this set of sequences has zero probability and thus the stopping time is a (non-defective) rv. We see from this that, although stopping rules are generally defined without the use of a probability measure, and the mapping from sample sequences to stopping nodes is similarly independent of the probability measure, the question of whether the stopping time is defective and whether it has moments is very dependent on the probability measure. £ § Theorem 7.2 (Wald’s identity). Let {Xi ; i ≥ 1} be IID and let ∞ (r) = ln{E erX } be the semi-invariant moment generating function of each Xi . Assume ∞ (r) is finite in an open interval (r− , r+ ) with r− < 0 < r+ . For each n ≥ 1, let Sn = X1 + · · · + Xn . Let α > 0 and β < 0 be arbitrary real numbers, and let N be the smal lest n for which either Sn ≥ α or Sn ≤ β . Then for al l r ∈ (r− , r+ ), E [exp(rSN − N ∞ (r))] = 1. (7.24) We first show how to use and interpret this theorem, and then prove it. The proof is quite simple, but will mean more after understanding the surprising power of this result. Wald’s identity can be thought of as a generating function form of Wald’s equality as established in Theorem 3.3. First note that the trial N at which a threshold is crossed in the theorem is a stopping time in the terminology of Chapter 3. Also, if we take the derivative with respect to r of both sides of (7.24), we get £ § E [SN − N ∞ 0 (r) exp{rSN − N ∞ (r)} = 0. Setting r = 0 and recalling that ∞ (0) = 0 and ∞ 0 (0) = X , this becomes Wald’s equality, E [SN ] = E [N ] X . (7.25) Note that this derivation of Wald’s equality is restricted to a random walk with two thresholds (and this automatically satisfies the constraint in Wald’s equality that E [N ] < 1). The result in Chapter 3 was more general, applyin...
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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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