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Unformatted text preview: MATH 507a FINAL EXAM SOLUTIONS Fall 2007 Prof. Alexander (1) ϕ S n /n ( t ) = Ee itS n /n = ϕ S n t n = ϕ t n n . For fixed t , as n → ∞ , ϕ t n n = 1 + ϕ (0) t n + o 1 n n → e ϕ (0) t = e iat , so S n /n → a in distribution, hence also in probability. (2)(a) P ( X n > c (log n ) α ) = e ac 2 (log n ) 2 α . (1) If α > 1 / 2 then 2 α > 1 so for large n , the exponent in (1) is > 2 log n , so P ( X n > c (log n ) α ) ≤ e 2 log n = 1 n 2 , so ∑ n P ( X n > c (log n ) α ) < ∞ so P ( X n > c (log n ) α i.o.) = 0. If α < 1 / 2 then 2 α < 1 so for large n then exponent in (1) is < 1 2 log n and P ( X n > c (log n ) α ) ≥ e 1 2 log n = 1 √ n , so ∑ n P ( X n > c (log n ) α ) = ∞ so P ( X n > c (log n ) α i.o.) = 1. If α = 1 / 2 then P ( X n > c (log n ) α ) = e ac 2 log n = n ac 2 , so X n P ( X n > c (log n ) α ) < ∞ ⇐⇒ ac 2 > 1 ⇐⇒ c > 1 √ a ....
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This note was uploaded on 06/23/2008 for the course MATH 507A taught by Professor Alexander during the Fall '08 term at USC.
 Fall '08
 Alexander
 Probability

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