Lec11[1] - Stat 150 Stochastic Processes Spring 2009...

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Stat 150 Stochastic Processes Spring 2009 Lecture 11: Return times for random walk Lecturer: Jim Pitman 1 Recurrence/Transience From last time: look at the simple random walk on Z with p q . S n := a + X 1 + ··· + X n . Compute: u 2 n = P 0 ( S 2 n = 0) f 2 n = P 0 ( T 0 = 2 n ) T 0 = first return time to 0 = inf { n : S n = 0 } (= if no such n exists) We already know that P 0 ( T 0 < ) = 1 if p = q = 1 2 p · q p + q · 1 = 2 q if p > q q · p q + p · 1 = 2 p if p < q To see this, P 1 ( T 0 < ) = ( q p if p q 1 if p < q let P 1 ( x < ) = x , then x = q · 1 + p · x 2 = x = 1 or q/p . x x 2 1 0 1
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Lecture 11: Return times for random walk 2 Recurrence/Transience We know u 2 n = ± 2 n n ² ( pq ) n What is f 2 n ? Let N 0 = n =0 1 ( S n = 0) = # of visits of sums to 0. From last class, E 0 N 0 = X n =0 ± 2 n n ² (1 / 2) 2 n (4 pq ) n ± 2 n n ² (1 / 2) 2 n c n (4 pq ) n = ( 1 if p = q < 1 if p 6 = q then E 0 N 0 = ( if p = q < if p 6 = q In fact, P 0 ( N 0 = ) = ( 1 for p = q 0 for p 6 = q . Obviously, E 0 N 0 < = P 0 ( N 0 < ) = 1, but E 0 N 0 = does not so obviously imply P 0 ( N 0 = ) = 1.
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Lec11[1] - Stat 150 Stochastic Processes Spring 2009...

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