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# a.home - Probability II MAP6473 4810 Home-A Prof JLF King...

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Probability II MAP6473 4810 Home-A Prof. JLF King 4May2010 Please. General instructions/notation are on the Checklist and “Conditional probability & conditional measures”. Note: MG means Martingale , and S T means Stopping Time . A1: Consider a MG ~ Y with pointwise bounds | Y 0 | 6 7 and n : | Y n +1 - Y n | 6 7 . 1 : Suppose that β is a S T with E ( β ) < . Prove that Y β is integrable and E ( Y β ) = E ( Y 0 ). A2: Consider an independent random-walk on the integers, where each step-probability depends on both position and time. A 3-spread D () is a mean-zero random vari- able with support on J := [ 3 .. 3 ] . That is, X j J P ( D = j ) = 1 and 2 : E ( D ) note === X j J j · P ( D = j ) = 0 . 3 : For each time n Z + and each position p Z , suppose that we have a 3-spread D n,p , and all these random variables are mutually independent. Then our random-walk is ~ S , where S 0 0 ( we start at the origin ) and S n +1 := S n + D n +1 , S n . Let τ () be the stopping time where the random- walk first hits position “5”. Use the preceding problem to prove that E ( τ ) must be infinite. [ Hint:
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