Question: Question 8 Let (Xn)neNo be a sequence of bounded random variables and consider the filtration (Fn)nen, defined by Fn := o(Xo, X1, . . .,

Question 8 Let (Xn)neNo be a sequence of bounded random variables and consider the filtration (Fn)nen, defined by Fn := o(Xo, X1, . . ., Xn), n E No. Assume that for each n E No we have EXn+1 |Fn] = 2Xn+ 1. Moreover, consider the process Zn : = 2 "Xn +2", neNo. Then, the process (Zn) neNo is a martingale with respect to (Fn) neNo. YES NO Question 9 Let (Xn)nen, be a branching process defined as in (8.1.1) in the script on page 310 which satisfies all the following: . Xo = 1, . One has limn- E[X ] = co. Then its extinction probability is equal to zero. YES NO

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