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assignment9 - Solutions to Homework 9 6.262 Discrete...

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Solutions to Homework 9 6.262 Discrete Stochastic Processes MIT, Spring 2011 Exercise 5.6 : Let { X n ; n 0 } be a branching process with = 1. X 0 Let Y , σ 2 be the mean and variance of the number of offspring of an individual. a) Argue that lim n X n exists with probability 1 and either has the value 0 (with →∞ probability F 10 ( )) or the value (with probability 1 F 10 ( )). Solution 5.6a We consider 2 special, rather trivial, cases before considering the impor- tant case (the case covered in the text). Let p i be the PMF of the number of offspring of each individual. Then if p 1 = 1, we see that X n = 1 for all n , so the statement to be argued is simply false. It is curious that this exercise has been given many times over the years with no one pointing this out. The next special case is where p 0 = 0 and p 1 < 1. Then X n +1 X n ( i.e. , the population never shrinks but can grow). Since X n ( ω ) is nondecreasing for each sample path, either lim n X n ( ω ) = or lim n X n ( ω ) = j for some j < . The latter case is impossible, →∞ →∞ since mj P jj = j p 1 and thus P m jj = p 1 0. Ruling out these two trivial cases, we have p 0 > 0 and p 1 < 1 p 0 . In this case, state 0 is recurrent ( i.e. , it is a trapping state) and states 1 , 2 ,..., are in a transient class. To see this, note that P 10 = p 0 > 0, so F 11 ( ) 1 p 0 < 1, which means by de±nition that state 1 is transient. All states i > 1 communicate with state 1, so by Theorem 5.1.1, all states j 1 are transient. Thus one can argue that the process has ‘no place to go’ other than 0 or . The following ugly analysis makes this precise. Note from Lemma 5.1.1 part 4 that lim P t jj = . t →∞ n t Since this sum is nondecreasing in t , the limit must exist and the limit must be ±nite This means that lim P n jj = 0 ± t n t →∞ Now we can write P n 1 j = n f P j n n 1 j j , from which it can be seen that lim t P = →∞ n t 1 j 0. From this, we see that for every ±nite integer , ± lim P n 1 j = 0 t →∞ n t j =1 This says that for every ± > 0, there is a t sufficiently large that the probability of ever entering states 1 to on or after step t is less than ± . Since ± > 0 is arbitrary, all sample paths (other than a set of probability 0) never enter states 1 to after some ±nite time. 1
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