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m507b-hw6soln-s08

m507b-hw6soln-s08 - MATH 507b ASSIGNMENT 6 SOLUTIONS SPRING...

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MATH 507b ASSIGNMENT 6 SOLUTIONS SPRING 2008 Prof. Alexander Chapter 7: (1.3) Let ξ be N (0 , t ), so Δ m,n has the distribution of 2 - n/ 2 ξ . We have 2 = t, E ( ξ 2 - t ) 2 = 2 t 2 , so the r.v.’s Δ 2 m,n - t 2 n are i.i.d. with mean 0. Therefore E 2 n X m =1 Δ 2 m,n - t ! 2 = E 2 n X m =1 Δ 2 m,n - t 2 n ! 2 (1) = 2 n X m =1 E Δ 2 m,n - t 2 n 2 = 2 n · 1 2 2 n E ( ξ 2 - t ) 2 = 2 t 2 2 n . Combining this with Chebyshev’s Inequality we obtain X n =1 P 2 n X m =1 Δ 2 m,n - t > ! X n =1 2 t 2 2 n 2 < , so by Borel-Cantelli, P 2 n X m =1 Δ 2 m,n - t > i.o. in n ! = 0 . Since is arbitrary this means 2 n m =1 Δ 2 m,n t a.s. (2.1) We have R = T 0 θ 1 + 1 so by the Markov property, P x ( R > 1 + t | F 1 ) = P x ( T 0 θ 1 > t | F 1 ) = P X 1 ( T 0 > t ) . Hence P x ( R > 1 + t ) = E x P X 1 ( T 0 > t ) = Z R p 1 ( x, y ) P y ( T 0 > t ) dy, since X 1 has the density p 1 ( x, y ) (as a function of y ) under P x . (2.3) By Theorem 2.6 (and its symmetric analog for “ B t < 0”), B t goes back and forth between positive and negative values infinitely often a.s. as t & 0. This means that almost surely, there exist s 1 > t 1 > s 2 > t 2 > . . . with B
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