4106-08-h7 - converge? 5. Consider the BLM, with S = 1 and...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
HMWK 7 1. X n = Y 1 × Y 2 ×···× Y n where X 0 = 1 and the Y i are iid with P ( Y = 0 . 5) = P ( Y = 1) = P ( Y = 1 . 5) = 1 / 3. Argue that X n converges and find the limit. 2. Consider the Gambler’s ruin Markov chain { X n } on { 0 , 1 , 2 , 3 , 4 } ( N = 4) with transition matrix P = 1 0 0 0 0 1 / 2 0 1 / 2 0 0 0 1 / 2 0 1 / 2 0 0 0 1 / 2 0 1 / 2 0 0 0 0 1 Suppose that X 0 = 1. Does X n converge wp1? Find the limiting rv X if so. Repeat for each initial condition X 0 = i, i = 0 , 1 , 2 , 3 , 4. 3. Continuation: Consider the Markov chain { X n } on { 0 , 1 , 2 , 3 , 4 } with transition matrix P = 1 0 0 0 0 2 / 3 0 0 1 / 3 0 1 / 5 1 / 5 1 / 5 1 / 5 1 / 5 0 1 / 2 0 1 / 2 0 0 0 0 1 Does X n converge wp1? Find the limiting rv X if so (it depends on initial conditions). 4. At time 0 an urn contains one red ball and one blue ball. At each time n after ( n = 1 , 2 , 3 ,... ) a ball is chosen at random from the urn and put back in together with one more ball of the same color. (So at time n there are n + 2 balls in the urn.) Let X n denote the proportion of balls at time n that are red. Show that { X n } is a MG. Does it
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 innitely 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...
View Full Document

This note was uploaded on 10/20/2010 for the course IEOR 4106 taught by Professor Whitt during the Fall '08 term at Columbia.

Ask a homework question - tutors are online