Discrete-time stochastic processes

18 eq 317 says that for non arithmetic distributions

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Unformatted text preview: number of trials required when the number is chosen in advance. Another example occurs in tree searches where a path is explored until further extensions of the path appear to be unprofitable. The first careful study of experimental situations where the number of trials depends on the data was made by the statistician Abraham Wald and led to the field of sequential analysis. Wald’s equality, in the next subsection, is quite simple but crucial to the study of these situations. Wald’s equality will be used again, along with a generating function equality known as Wald’s identity, when we study random walks. An important part of experiments that stop after a random number of trials is the rule for stopping. Such a rule must specify, for each sample function, the trial at which the experiment stops, i.e., the final trial after which no more trials are performed. Thus the rule for stopping must specify a positive, integer valued, random variable J , called the stopping time, mapping sample functions into the trial at which the experiment stops. We still view the sample space as the set of sample value sequences for the never ending sequence of random variables X1 , X2 , . . . . That is, even if the experiment is stopped at the 3.3. EXPECTED NUMBER OF RENEWALS 101 end of the second trial, we still visualize the 3rd, 4th, . . . random variables as having sample values as part of the sample function. In other words, we visualize that the experiment continues forever, but that the observer stops watching at the end of the stopping point. From the standpoint of applications, it doesn’t make any difference whether the experiment continues or not after the observer stops watching. From a mathematical standpoint, however, it is far preferable to view the experiment as continuing so as to avoid confusion and ambiguity about what it means for the variables X1 , X2 , . . . to be IID when the very existence of later variables depends on earlier sample values. The intuitive notion of stopping includes the notion that a decision to stop before trial n should depend only on the results before trial n. In other words, we want to exclude from stopping rules those rules that allow the experimenter to peek at subsequent values before making the decision to stop or not.3 In other words, the event {J ≥ n}, i.e., the event that the nth experiment is performed, should be independent of Xn and all subsequent trials. More precisely, Definition 3.1. A stopping time4 J for a sequence of rv’s X1 , X2 , . . . , is a positive integer valued rv such that for each n ≥ 1, the event {J ≥ n} is statistical ly independent of (Xn , Xn+1 , . . . ). It is convenient in working with stopping rules to use an indicator function In for the event {J ≥ n} for each n ≥ 1. That is, In = 1 if J ≥ n and In = 0 otherwise. A stopping time is then a positive integer-valued rv for which each associated indicator function In is independent of Xn , Xn+1 , . . . for each n. Thus the rv In is a binary rv that takes the value 1 if the experiment includes the nth trial, and the value 0 otherwise. Since we assume that the first observation is always made (i.e., J is a positive random variable), I1 = 1 with probability 1. We can view In as a decision rule exercised by an observer to determine whether to continue with the nth trial. In many applications, however, including that of establishing the elementary renewal theorem, there is no real notion of an observer, but only of the specification of some condition, not involving peeking, that is met after some random number of trials. Since J ≥ n implies that J ≥ j for all j < n, the indicator functions have the corresponding property that In = 1 implies that Ij = 1 for j < n. Also, since J is a rv, and thus finite with probability 1, limn→1 Pr {In = 1} must be 0. Thus, according to the definition, stopping must take place eventually, although not necessarily with any finite bound. We see that each decision rule In is a function of the stopping time J , and the stopping time J is also determined by all the decision rules (see Exercise 3.4). The notion that a stopping rule should not allow peeking then means, for each n > 1, that In , the decision whether or not to observe Xn , should depend only on X1 , . . . , Xn−1 . 3 For example, poker players do not take kindly to a player who bets on a hand and then withdraws his bet when someone else wins the hand. 4 Stopping times are sometimes called optional stopping times. 102 3.3.3 CHAPTER 3. RENEWAL PROCESSES Wald’s equality An important question that arises with stopping rules is to evaluate the sum SJ of the P random variables up to the stopping time, i.e., SJ = J =1 Xn . Many gambling strategies n and investing strategies involve some sort of rule for when to stop, and it is important to understand the rv SJ . Wald’s equality is very useful in establishing a very simple way to find E [SJ ]. Theorem 3.3 (Wald’s Equality). Let {Xn ; n ≥ 1} be IID rv’s, e...
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