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Unformatted text preview: Lecture XIV Let ϖ represent the entire random sequence { Z t }. As discussed last time, our interest typically centers around the averages of this sequence: ( 29 ∑ = = n t t n Z n b 1 1 ω Definition 2.9: Let { b n ( ϖ )} be a sequence of realvalued random variables. We say that b n ( ϖ ) converges almost surely to b , written if and only if there exists a real number b such that ( 29 b b s a n → . . ω ( 29 [ ] 1 : = → b b P n ω ω The probability measure P describes the distribution of ϖ and determines the joint distribution function for the entire sequence { Z t }.  Other common terminology is that b n ( ϖ ) converges to b with probability 1 (w.p.1) or that b n ( ϖ ) is strongly consistent for b . Example 2.10: Let where { Z t } is a sequence of independently and identically distributed (i.i.d.) random variables with E ( Z t )= μ < ∞ . Then by the Komolgorov strong law of large numbers (Theorem 3.1). ∑ = = n t t n Z n Z 1 1 μ → . . s a n Z Proposition 2.11: Given g : R k → R l ( k , l <∞) and any sequence { b n } such that where b n and b are k x 1 vectors, if g is continuous at b , then b b s a n → . . ( 29 ( 29 b g b g s a n → . . Theorem 2.12: Suppose y = X β + ε ; X ’ ε / n → a.s. 0; X ’ X / → a.s. M , finite and positive definite.  Then β n exists a.s. for all n sufficiently large, and β n → a.s. β . Proof: Since X ’ X / n → a.s. M , it follows from Proposition 2.11 that det( X ’ X / n ) → a.s. det( M ). Because M is positive definite by (iii), det( M )>0. It follows that det( X ’ X / n )>0 a.s. for all n sufficiently large, so ( X ’ X / n )1 exists a.s. for all n sufficiently large. Hence ( 29 n y X n X X n ' ' ˆ 1 ≡ β In addition,...
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This note was uploaded on 11/08/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.
 Fall '09
 CARRIKER

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