Unformatted text preview: converge? 5. Consider the BLM, with S = 1 and S n = Y 1 ··· Y n , n ≥ 1 where d = 0 . 5, u = 1 . 5 and p = 0 . 50. (a) Show that E ( S n ) = 1 , n ≥ 0, but wp1, S n → 0 as n → ∞ . (b) Let ± > 0 be very small. Let us change u to be (1 . 5)(1 + ± ) so that it is strictly larger than 1 . 5. We keep all other parameters the same as before. Show now that E ( S n ) → ∞ , and that if ± is chosen small enough it still remains true that S n → 0. On average you will become inﬁnitely rich, but with certainty you will go broke! (Interesting?) (c) Continuation: Under the conditions in (b), show that { S n } while not a MG is in fact a SUBMG. 1...
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 Fall '08
 Whitt
 Operations Research, Markov chain, Yi, Xn

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