Lecture 15-2007 - Lecture XIV Let represent the entire...

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Unformatted text preview: Lecture XIV Let 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 Definition 2.9: Let { b n ( )} be a sequence of real-valued random variables. We say that b n ( ) converges almost surely to b , written if and only if there exists a real number b such that ( 29 b b s a n . . ( 29 [ ] 1 : = b b P n The probability measure P describes the distribution of and determines the joint distribution function for the entire sequence { Z t }.  Other common terminology is that b n ( ) converges to b with probability 1 (w.p.1) or that b n ( ) 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 )= < . Then by the Komolgorov strong law of large numbers (Theorem 3.1). = = n t t n Z n Z 1 1 . . 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 + ; X / n a.s. 0; X X / a.s. M , finite and positive definite.  Then n exists a.s. for all n sufficiently large, and n a.s. . 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 ' ' 1- In addition,...
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Lecture 15-2007 - Lecture XIV Let represent the entire...

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