sol9_f08 - Math 136 Stochastic Processes Suggested...

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Unformatted text preview: Math 136 - Stochastic Processes Suggested Excercises, Autumn 2008 Questions? See Bo Shen. 1. Exercise 4.6.8. Suppose { Z n } is a branching process with P ( N = 1) < 1 and Z = 1. Show that P ( lim n →∞ Z n = ∞ ) = 1- p ex , first in case m ≤ 1, then in case P ( N = 0) = 0 and finally using the preceding exercise, for m > 1 and P ( N = 0) > 0. ANS: Since P ( N = 1) < 1 we have by Propositions 4.6.3 and 4.6.5 that p ex = 1 when m ≤ 1. That is, in this case w.p.1. Z n = 0 for all n large enough, yielding the stated claim. In contrast, if P ( N = 0) = 0 then Z n is non-decreasing, so p ex = 0. Further, in this case Z n is bounded only if N ( k ) 1 = 1 for all k large enough, which with P ( N = 1) < 1 occurs with zero probability, again resulting with the stated claim. Finally, for m > 1 and P ( N = 0) > 0 we have from Exercise 4.6.7 that p ex = ρ ∈ (0 , 1) and further, 1- p ex = 1- P ( M ∞ = 1) = P ( M ∞ = 0) = P ( lim n →∞ ρ Z n = 0) = P ( lim n →∞ Z n = ∞ ) , as claimed. 2. Exercise 4.6.9. Let { Z n } be a branching process with Z = 1. Compute p ex in each of the following situations and specify for which values of the various parameters the extinction is certain. (a) The offspring distribution satisfies, for some 0 < p < 1, P ( N = 0) = p, P ( N = 2) = 1- p. ANS: We have that m = E N = 2(1- p ) with m ≤ 1 if and only if p ≥ 1 / 2. Thus, p ex = 1 when p ≥ 1 / 2 by Proposition 4.6.3 (for p > 1 / 2) and Proposition 4.6.5 (for p = 1 / 2, applicable since here P ( N = 1) = 0 < 1). Finally, if p < 1 / 2 then m > 1 so { Z n } is super-critical with P ( N = 0) = p > 0. We have shown in Exercise 4.6.7 that in this case p ex is the unique solution in (0 , 1) of 0 = x- φ ( x ) = x- P ( N = 0)- P ( N = 2) x 2 = x- p- (1- p ) x 2 (taking the function φ ( x ) per equation (4.6.2) that corresponds to our law of N ). As x- p- (1- p ) x 2 = (1- p )(1- x )( x- p/ (1- p )) , we conclude that p ex = p/ (1- p ) < 1 when p < 1 / 2. 1 (b) The offspring distribution is (shifted) Geometric, i.e. for some 0 < p < 1, P ( N = k ) = p (1- p ) k , k = 0 , 1 , 2 ,... ANS: We have now that m = E N = ∑ ∞ k =1 kp (1- p ) k = (1- p ) /p (where to get the last identity differentiate in p the identity ∑ ∞ k =0 (1- p ) k = 1 /p and multiply both sides by- p (1- p )). As in part (a), if p ≥ 1 / 2 then m ≤ 1 and consequently p ex = 1 (for here too P ( N = 1) = p (1- p ) < 1). In contrast, p < 1 / 2 yields a super-critical branching process with P ( N = 0) = p > 0, so again from Exercise 4.6.7 we have that p ex is the unique solution in (0 , 1) of 0 = x- φ ( x ) = x- ∞ X k =0 P ( N = k ) x k = x- p ∞ X k =0 (1- p ) k x k = x- p 1- (1- p ) x ....
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sol9_f08 - Math 136 Stochastic Processes Suggested...

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